Induced subgraphs and tree decompositions
XV. Even-hole-free graphs with bounded clique number have logarithmic treewidth

Maria Chudnovsky^† , Peter Gartland^‡ , Sepehr Hajebi ^§ , Daniel Lokshtanov^‡ and Sophie Spirkl^§∥

Abstract.

We prove that for every integer $t\geq 1$ there exists an integer $c_{t}\geq 1$ such that every $n$ -vertex even-hole-free graph with no clique of size $t$ has treewidth at most $c_{t}\log{n}$ . This resolves a conjecture of Sintiari and Trotignon, who also proved that the logarithmic bound is asymptotically best possible. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and Coloring admit polynomial-time algorithms on this class of graphs. As a consequence, for every positive integer $r$ , $r$ -Coloring can be solved in polynomial time on even-hole-free graphs without any assumptions on clique size.

As part of the proof, we show that there is an integer $d$ such that every even-hole-free graph has a balanced separator which is contained in the (closed) neighborhood of at most $d$ vertices. This is of independent interest; for instance, it implies the existence of efficient approximation algorithms for certain NP-hard problems while restricted to the class of all even-hole-free graphs.

^‡ Department of Computer Science, University of California Santa Barbara, Santa Barbara, CA, USA. Supported by NSF grant CCF-2008838.

^§Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

^†Princeton University, Princeton, NJ, USA. Supported by NSF-EPSRC Grant DMS-2120644 and by AFOSR grant FA9550-22-1-0083.

^∥ We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence RGPIN-2020-03912]. This project was funded in part by the Government of Ontario. This research was completed while Spirkl was an Alfred P. Sloan fellow.

1. Introduction

All graphs in this paper are finite and simple. For a graph $G$ , we denote by $V(G)$ the vertex set of $G$ , and by $E(G)$ the edge set of $G$ . For graphs $H,G$ , we say that $H$ is an induced subgraph of $G$ if $V(H)\subseteq V(G)$ , and $x,y\in V(H)$ are adjacent in $H$ if and only if $xy\in E(G)$ . A hole in a graph is an induced cycle with at least four vertices. The length of a hole is the number of vertices in it. A hole is even it it has even length, and odd otherwise. The class of even-hole-free graphs has attracted much attention in the past due to its somewhat tractable, yet very rich structure (see the survey [37]). In addition to stuctural results, much effort was put into designing efficient algorithms for even-hole-free graphs (to solve problems that are NP-hard in general). This is discussed in [37], while [1, 12, 15, 28] provide examples of more recent work. Nevertheless, many open questions remain. Among them is the complexity of several algorithmic problems: Stable Set, Vertex Cover, Dominating Set, $k$ -Coloring and Coloring.

For a graph $G=(V(G),E(G))$ , a tree decomposition $(T,\chi)$ of $G$ consists of a tree $T$ and a map $\chi:V(T)\to 2^{V(G)}$ with the following properties:

(i)

For every vertex $v\in V(G)$ , there exists $t\in V(T)$ such that $v\in\chi(t)$ .
(ii)

For every edge $v_{1}v_{2}\in E(G)$ , there exists $t\in V(T)$ such that $v_{1},v_{2}\in\chi(t)$ .
(iii)

For every vertex $v\in V(G)$ , the subgraph of $T$ induced by $\{t\in V(T)\mid v\in\chi(t)\}$ is connected.

For each $t\in V(T)$ , we refer to $\chi(t)$ as a bag of $(T,\chi)$ . The width of a tree decomposition $(T,\chi)$ , denoted by $\operatorname{width}(T,\chi)$ , is $\max_{t\in V(T)}|\chi(t)|-1$ . The treewidth of $G$ , denoted by $\operatorname{tw}(G)$ , is the minimum width of a tree decomposition of $G$ .

Treewidth, first introduced by Robertson and Seymour in the context of graph minors, is an extensively studied graph parameter, mostly due to the fact that graphs of bounded treewidth exhibit interesting structural [35] and algorithmic [11] properties. It is easy to see that large complete graphs and large complete bipartite graphs have large treewidth, but there are others (in particular a subdivided $k\times k$ -wall, which is a planar graph with maximum degree three, and which we will not define here). A theorem of [35] characterizes precisely excluding which graphs as minors (and in fact as subgraphs) results in a class of bounded treewidth.

Bringing tree decompositions into the world of induced subgraphs is a relatively recent endeavor. It began in [36], where the authors observed yet another evidence of the structural complexity of the class of even-hole-free graphs: the fact that there exist even-hole-free graphs of arbitrarily large treewidth (even when large complete subgraphs are excluded). Closer examination of their constructions led the authors of [36] to make the following two conjectures (the diamond is the unique simple graph with four vertices and five edges):

Conjecture 1.1 (Sintiari and Trotignon [36]).

For every integer $t\geq 1$ , there exists a constant $c_{t}$ such that every even-hole-free graph $G$ with no induced diamond and no clique of size $t$ satisfies $\operatorname{tw}(G)\leq c_{t}$ .

Conjecture 1.2 (Sintiari and Trotignon [36]).

For every integer $t\geq 1$ , there exists a constant $C_{t}$ such that every even-hole-free graph $G$ with no clique of size $t$ satisfies $\operatorname{tw}(G)\leq C_{t}\log|V(G)|$ .

Conjecture 1.1 was resolved in [8]. Here we prove Conjecture 1.2, thus closing the first line of research on induced subgraphs and tree decompositions.

We remark that the construction of [36] shows that the logarithmic bound of Conjecture 1.2 is asymptotically best possible. Furthermore, our main result implies that many algorithmic problems that are NP-hard in general (among them that Stable Set, Vertex Cover, Dominating Set and Coloring) admit polynomial-time algorithms in the class of even-hole-free graphs with bounded clique number. As a consequence, for every positive integer $r$ , $r$ -Coloring can be solved in polynomial time on even-hole-free graphs without any assumptions on clique size.

Before we proceed, we introduce some notation and definitions. Let $G=(V(G),E(G))$ be a graph. For a set $X\subseteq V(G)$ , we denote by $G[X]$ the subgraph of $G$ induced by $X$ . For $X\subseteq V(G)$ , we denote by $G\setminus X$ the subgraph induced by $V(G)\setminus X$ . In this paper, we use induced subgraphs and their vertex sets interchangeably.

For graphs $G$ and $H$ , we say that $G$ contains $H$ if some induced subgraph of $G$ is isomorphic to $H$ . For a family $\mathcal{H}$ of graphs, $G$ contains $\mathcal{H}$ if $G$ contains a member of $\mathcal{H}$ , and we say that $G$ is $\mathcal{H}$ -free if $G$ does not contain $\mathcal{H}$ . A clique in a graph is a set of pairwise adjacent vertices, and a stable set is a set of vertices no two of which are adjacent. Let $v\in V(G)$ . The open neighborhood of $v$ , denoted by $N(v)$ , is the set of all vertices in $V(G)$ adjacent to $v$ . The closed neighborhood of $v$ , denoted by $N[v]$ , is $N(v)\cup\{v\}$ . Let $X\subseteq V(G)$ . The open neighborhood of $X$ , denoted by $N(X)$ , is the set of all vertices in $V(G)\setminus X$ with at least one neighbor in $X$ . The closed neighborhood of $X$ , denoted by $N[X]$ , is $N(X)\cup X$ . If $H$ is an induced subgraph of $G$ and $X\subseteq V(G)$ , then $N_{H}(X)=N(X)\cap H$ and $N_{H}[X]=N_{H}(X)\cup(X\cap H)$ . Let $Y\subseteq V(G)$ be disjoint from $X$ . We say $X$ is complete to $Y$ if all possible edges with an end in $X$ and an end in $Y$ are present in $G$ , and $X$ is anticomplete to $Y$ if there are no edges between $X$ and $Y$ .

Given a graph $G$ , a path in $G$ is an induced subgraph of $G$ that is a path. If $P$ is a path in $G$ , we write $P=p_{1}\hbox{-}\cdots\hbox{-}p_{k}$ to mean that $V(P)=\{p_{1},\dots,p_{k}\}$ , and $p_{i}$ is adjacent to $p_{j}$ if and only if $|i-j|=1$ . We call the vertices $p_{1}$ and $p_{k}$ the ends of $P$ , and say that $P$ is a path from $p_{1}$ to $p_{k}$ . The interior of $P$ , denoted by $P^{*}$ , is the set $V(P)\setminus\{p_{1},p_{k}\}$ . The length of a path $P$ is the number of edges in $P$ . We denote by $C_{k}$ a cycle with $k$ vertices.

Next, we describe a few types of graphs that we will need (see Figure 1).

Refer to caption — Figure 1. Theta, prism and an even wheel. Dashed lines represent paths of length at least one.

A theta is a graph consisting of three internally vertex-disjoint paths $P_{1}=a\hbox{-}\cdots\hbox{-}b$ , $P_{2}=a\hbox{-}\cdots\hbox{-}b$ , and $P_{3}=a\hbox{-}\cdots\hbox{-}b$ , each of length at least 2, such that $P_{1}^{*},P_{2}^{*},P_{3}^{*}$ are pairwise anticomplete. In this case we call $a$ and $b$ the ends of the theta.

A prism is a graph consisting of three vertex-disjoint paths $P_{1}=a_{1}\hbox{-}\cdots\hbox{-}b_{1}$ , $P_{2}=a_{2}\hbox{-}\cdots\hbox{-}b_{2}$ , and $P_{3}=a_{3}\hbox{-}\cdots\hbox{-}b_{3}$ , each of length at least 1, such that $a_{1}a_{2}a_{3}$ and $b_{1}b_{2}b_{3}$ are triangles, and no edges exist between the paths except those of the two triangles.

A pyramid is a graph consisting of three paths $P_{1}=a\hbox{-}\cdots\hbox{-}b_{1}$ , $P_{2}=a\hbox{-}\cdots\hbox{-}b_{2}$ , and $P_{3}=a\hbox{-}\cdots\hbox{-}b_{3}$ such that $P_{1}\setminus\{a\},P_{2}\setminus\{a\},P_{3}\setminus\{a\}$ are pairwise disjoint, at least two of the three paths $P_{1},P_{2},P_{3}$ have length at least two, $b_{1}b_{2}b_{3}$ is triangle (called the base of the pyramid), and no edges exist between $P_{1}\setminus\{a\},P_{2}\setminus\{a\},P_{3}\setminus\{a\}$ except those of the triangle $b_{1}b_{2}b_{3}$ . The vertex $a$ is called the apex of the pyramid.

A wheel $(H,x)$ in $G$ is a pair where $H$ is a hole and $x$ is a vertex with at least three neighbors in $H$ . A wheel $(H,x)$ is even if $x$ has an even number of neighbors on $H$ .

Let $\mathcal{C}$ be the class of ( $C_{4}$ , theta, prism, even wheel)-free graphs (sometimes called “ $C_{4}$ -free odd-signable” graphs). For every integer $t\geq 1$ , let $\mathcal{C}_{t}$ be the class of all graphs in $\mathcal{C}$ with no clique of size $t$ . It is easy to see that every even-hole-free graph is in $\mathcal{C}$ .

The reader may be familiar with [3] where a special case of Conjecture 1.2 was proved; moreover, only one Lemma of [3] uses the fact that the set-up there is not the most general case. At the time, the authors of [3] thought that the full proof of Conjecture 1.2 would follow the general outline of [3], fixing the one missing lemma. That is not what happened. The proof in the present paper takes a different path: while it relies on insights and a general understanding of the class of even-hole-free graphs gained in [3], and uses several theorems proved there, a significant detour is needed.

The first part of the paper is not concerned with treewidth at all. Instead, we focus on the following question: given two non-adjacent vertices in a graph in $\mathcal{C}$ of bounded clique number, how many internally vertex-disjoint paths can there be between them? Given that if instead of “internally vertex-disjoint” we say “with pairwise anticomplete interiors”, then the answer is “two”, this is somewhat related to the recent work on the induced Menger theorem [21, 25, 31]. Let $k\geq 1$ be an integer and let $a,b\in V(G)$ . We say that $ab$ is a $k$ -banana if $a$ is non-adjacent to $b$ , and there exist $k$ pairwise internally-vertex-disjoint paths from $a$ to $b$ . Note that if $ab$ is a $k$ -banana in $G$ , then $ab$ is an $l$ -banana in $G$ for every $l\leq k$ . We prove:

Theorem 1.3.

For every integer $t\geq 1$ , there exists a constant $c_{t}$ such that if $G\in\mathcal{C}_{t}$ , then $G$ contains no $c_{t}\log|V(G)|$ -banana.

The next step in the proof of Conjecture 1.2 is the following. Let $c\in[0,1]$ and let $w$ be a normal weight function on $G$ , that is, $w:V(G)\rightarrow\mathbb{R}_{\geq 0}$ satisfies $\sum_{v\in V(G)}w(v)=1$ . A set $X\subseteq V(G)$ is a $(w,c)$ -balanced separator if $w(D)\leq c$ for every component $D$ of $G\setminus X$ . The set $X$ is a $w$ -balanced separator if $X$ is a $(w,\frac{1}{2})$ -balanced separator. We show:

Theorem 1.4.

There is an integer $d$ with the following property. Let $G\in\mathcal{C}$ and let $w$ be a normal weight function on $G$ . Then there exists $Y\subseteq V(G)$ such that

•

$|Y|\leq d$ , and
•

$N[Y]$ is a $w$ -balanced separator in $G$ .

We then use Theorem 1.3 and Theorem 1.4 to prove our main result:

Theorem 1.5.

For every integer $t\geq 1$ , there exists a constant $c_{t}$ such that every $G\in\mathcal{C}_{t}$ satisfies $\operatorname{tw}(G)\leq c_{t}\log|V(G)|$ .

1.1. Proof outline and organization

We only include a few definitions in this section; all the necessary definitions appear in later parts of the paper. Our first goal is to prove Theorem 1.3. Let $a,b\in V(G)$ be non-adjacent. Recall that a separation of a graph $G$ is a triple $(X,Y,Z)$ of pairwise disjoint subsets of $G$ with $X\cup Y\cup Z=G$ such that $X$ is anticomplete to $Z$ . Similarly to [6], we use the fact that graphs in $\mathcal{C}_{t}$ admit a natural family of separations that correspond to special vertices of the graph called “hubs” and are discussed in Section 3. Unfortunately, the interactions between these separations may be complex, and, similarly to [5], we use degeneracy to partition the set of all hubs other than $a,b$ (which yields a partition of all the natural separations) of an $n$ -vertex graph $G$ in $\mathcal{C}_{t}$ into collections $S_{1},\dots,S_{p}$ , where each $S_{i}$ is “non-crossing” (this property is captured in Lemma 4.9), $p\leq C(t)\log n$ (where $C(t)$ only depends on $t$ and works for all $G\in\mathcal{C}_{t}$ ) and each vertex of $S_{i}$ has at most $d$ (where $d$ depends on $t$ ) neighbors in $\bigcup_{j=i}^{p}S_{j}$ . We prove a strengthening of Theorem 1.3, which asserts that the size of the largest $ab$ -banana is bounded by a linear function of $p$ .

We observe that a result of [3] implies that there exists a an integer $k$ (that depends on $t$ ) such that if $G\in\mathcal{C}_{t}$ has no hubs other than $a,b$ , then $ab$ is not a $k$ -banana in $G$ . More precisely, the result of [3] states that if $a$ and $b$ are joined by $k$ internally-vertex-disjoint paths $P_{1},\dots,P_{k}$ , then for at least one $i\in\{1,\dots,k\}$ , the neighbor of $a$ in $P_{i}$ is a hub.

We now proceed as follows. Let $m=2d+k$ (where $d$ comes from the degeneracy partition and $k$ from the previous paragraph); suppose that $ab$ is a $(4m+2)(m-1)$ -banana in $G$ and let $\mathcal{P}$ be the set of all paths of $G$ with ends $a,b$ .

Let $S_{1},\dots,S_{p}$ denote the partition of hubs as decribed above. We proceed by induction on $p$ and describe a process below that finds an induced subgraph $H$ of $G$ in which:

•

Vertices in $S_{1}$ are no longer hubs;
•

If there are not many internally disjoint $ab$ -paths in $H$ , we can “lift” this to $G$ .

We first consider a so-called $m$ -lean tree decomposition $(T,\chi)$ of $G$ (discussed in Section 2). By examining the intersection graphs of subtrees of a tree we deduce that there exist two (not necessarily distinct) vertices $t_{1},t_{2}\in V(T)$ such that for every $ab$ path $P$ , the interior of $P$ meets $\chi(t_{1})\cup\chi(t_{2})$ . We also argue that $a,b\in\chi(t_{1})\cup\chi(t_{2})$ . A vertex $u$ of $S_{1}$ is bad if $u$ has large degree (at least $D=2m(m-1)$ ) in the torso of $\chi(t_{1})$ , or $u$ has large degree in the torso of $\chi(t_{2})$ , or $u$ is adjacent to both $a$ and $b$ . We show that there are at most three bad vertices in $S_{1}$ .

We would like to bound the size of the largest $ab$ -banana in the union of the two torsos. Unfortunately, the torso of $\chi(t_{i})$ may not be a graph in $\mathcal{C}_{t}$ . Instead, we find an induced subgraph of $G$ , which we call $\beta$ , that consists of $\chi(t_{1})\cup\chi(t_{2})$ together with a collection of disjoint vertex sets $\operatorname{Conn}(t_{i},t)$ for $t\in V(T)\setminus\{t_{1},t_{2}\}$ and $i\in\{1,2\}$ except vertices $t$ on the $t_{1}\hbox{-}t_{2}$ -path in $T$ , where each $\operatorname{Conn}(t_{i},t)$ “remembers” the component of $G\setminus(\chi(t_{1})\cup\chi(t_{2}))$ that meets $\chi(t)$ . Importantly, no vertex of $\beta\setminus(\chi(t_{1})\cup\chi(t_{2}))$ is a hub of $\beta$ , and all but at most three vertices of $S_{1}$ have bounded degree in the union of the torso of $\chi(t_{1})$ and the torso of $\chi(t_{2})$ .

We next decompose $\beta$ , simultaneously, by all the separations corresponding to the hubs in $S_{1}$ that are not bad, and delete the set of (at most three) bad vertices of $S_{1}$ . We denote the resulting graph by $\beta^{A}(S_{1})$ and call it the “central bag” for $S_{1}$ . The parameter $p$ is smaller for $\beta^{A}(S_{1})$ than it is for $G$ , and so we can use induction to obtain a bound on the largest size of an $ab$ -banana in $\beta^{A}(S_{1})$ . Since our goal is to bound the size of an $ab$ -banana in $G$ by a linear function of $p$ , we now need to show that the size of the largest $ab$ -banana does not grow by more than an additive constant when we move from $\beta^{A}(S_{1})$ to $G$ .

We start with a smallest subset $X$ of $\beta^{A}(S_{1})$ that separates $a$ from $b$ in $\beta^{A}(S_{1})$ (and whose size is bounded by Menger’s theorem) and show how to transform in into a set $Y$ separating $a$ from $b$ in $\beta$ , while increasing the size of its intersection with $\chi(t_{1})\cup\chi(t_{2})$ by at most an additive constant, and ensuring the number of sets $\operatorname{Conn}(t_{i},t)$ that $Y$ meets is bounded by a constant. Then we repeat a similar procedure to obtain a set $Z$ of vertices separating $a$ from $b$ in $G$ , making sure that the increase in size is again bounded by an additive constant.

Let us now discuss how we obtain the bound on the growth of the separator. In the first step, to obtain $Y$ , we add to $X$ the neighbor sets of the vertices of $S_{1}\cap X$ . Since while constructing $\beta^{A}(S_{1})$ we have deleted all the bad vertices of $S_{1}$ , the number of vertices of $\chi(t_{1})\cup\chi(t_{2})$ added to $X$ is at most $2|S_{1}\cap X|D$ , and $Y\setminus X$ meets at most $2|S_{1}\cap X|D$ of the sets $\operatorname{Conn}(t_{i},t)$ . Note that the bound on the size of $X$ that we have depends on $p$ , which may be close to $\log|V(G)|$ , so another argument is needed to obtain a constant bound on $|S_{1}\cap X|$ and on the number of sets $\operatorname{Conn}(t_{i},t)$ that meet $Y$ . This is a consequence of Theorem 6.3, because no vertex of $S_{1}$ is a hub in $\beta^{A}(S_{1})$ (this is proved in Theorem 4.10), and no vertex of $\operatorname{Conn}(t_{i},t)$ is a hub in $\beta$ . (The proof of Theorem 6.3 analyzes the structure of minimal separators in graphs in $\mathcal{C}$ using tools developed in Section 5.)

In the second growing step, we start with the set $Y$ obtained in the previous paragraph. Then we add to $Y$ the following subsets (here we describe the cases when $t_{1}\neq t_{2}$ ; if $t_{1}=t_{2}$ the argument is similar).

(1)

$\chi(t_{1})\cap\chi(t_{1}^{\prime})$ where $t_{1}^{\prime}$ is the unique neighbor of $t_{1}$ in the path in $T$ from $t_{1}$ to $t_{2}$ .
(2)

$\chi(t_{2})\cap\chi(t_{2}^{\prime})$ where $t_{2}^{\prime}$ is the unique neighbor of $t_{2}$ in the path in $T$ from $t_{1}$ to $t_{2}$ .
(3)

$\chi(t_{1})\cap\chi(t)$ for every $t\in N(t_{1})\setminus\{t_{1}^{\prime}\}$ such that $Y\cap\operatorname{Conn}(t_{1},t)\neq\emptyset$ .
(4)

$\chi(t_{2})\cap\chi(t)$ for every $t\in N(t_{2})\setminus\{t_{2}^{\prime}\}$ such that $Y\cap\operatorname{Conn}(t_{2},t)\neq\emptyset$ .
(5)

The set of all bad vertices of $S_{1}$ .

One of the properties of $m$ -lean tree decompositions is that the size of each adhesion (intersection of “neighboring” bags) is less than $m$ . The number of adhesions added to $Y$ is again bounded since the number of distinct sets $\operatorname{Conn}(t_{i},t)$ that meet $Y$ is bounded. This completes the proof of Theorem 1.3.

The next key ingredient in the proof of Theorem 1.5 is Theorem 1.4, asserting that there is an integer $C$ such that for every graph $G\in\mathcal{C}$ and every normal weight function $w$ on $G$ , there is a $w$ -balanced separator $X$ in $G$ such that $X$ is contained in the union of the neighborhoods of at most $C$ vertices of $G$ . To prove that, we first prove a decomposition theorem, similar to the one giving us the natural separations associated with hubs, but this time the separations come from pyramids in $G$ . We then use the two kinds of separations (the kind that come from hubs and the kind that come from pyramids) as follows.

For $X\subseteq V(G)$ we say that a set $Y\subseteq V(G)$ dominates $X$ if $X\subseteq N[Y]$ . By Lemma 8.5 (from [14]) there is a path $P=p_{1}\hbox{-}\cdots\hbox{-}p_{k}$ in $G$ such that $w(D)\leq\frac{1}{2}$ for every component $D$ of $G\setminus N[P]$ . We now use a “sliding window” argument: we divide $P$ into subpaths, each with the property that its neighborhood can be dominated by a small, but not too small, set of vertices. Using the decomposition theorems above, we find a bounded-size set $X(W)$ such that, except for our window $W$ , the path $P$ is disjoint from $N[X(W)]$ , and $N[X(W)]$ separates the subpath of $P$ before our current window from that after our current window. This means (unless $N[X]$ is a balanced separator) that the big component of $G\setminus N[X]$ does not contain either the subpath of $P$ before our window, or the subpath after our window. By looking at the point in the path where this answer changes from “before” to ”after” (and showing that such a point exists), and by combining the separators for the two windows before and after this point as well as small dominating sets for the neighborhood of those two windows, we are able to find a $w$ -balanced separator with bounded domination number. This completes the proof of Theorem 1.4. We remark that Theorem 1.4 applies to all graphs in $\mathcal{C}$ and does not assume a bound on the clique number.

The next step in the proof of Theorem 1.5 is to prove Theorem 9.2, asserting that for every integer $L$ , if a graph $G$ contains no $L$ -banana, then for every normal weight function $w$ on $G$ , if $G$ has a $w$ -balanced separator $N[X]$ , then $G$ has an $w$ -balanced separator $Y$ and a clique $T\subseteq X$ such that $|Y\setminus T|\leq 3|X|L$ . This step uses $3L$ -lean tree decompositions of $G$ and works for all graphs $G$ .

Now Theorem 9.2 and Theorem 1.3 together imply that for every $t$ , there exists an integer $q$ , depending on $t$ , such that for every $G\in\mathcal{C}_{t}$ , $G$ has a balanced separator of size at most $q\log|V(G)|+t$ ; that is Theorem 9.1. By Lemma 10.1 this immediately implies the desired bound on the treewidth of $G$ .

The paper is organized as follows. In Section 2, we discuss lean tree decompositions and their properties, along with other classical results in graph theory. For some of them we prove variations tailored specifically to our needs. In Section 3 we summarize results guaranteeing the existence of separations associated with hubs. We also establish a stronger version of Theorem 1.3 for the case when the set of hubs in $G$ is very restricted. In Section 4 we discuss the construction of the graphs $\beta$ and $\beta^{A}(S_{1})$ , and how to use $ab$ -separators there to obtains $ab$ -separators in $G$ . In Section 5 we analyze the structure of minimal separators in graphs of $\mathcal{C}$ . In Section 6 we use the results of Section 5 to obtain a bound on the size of a stable set of non-hubs in an $ab$ -separator of smallest size. Section 7 puts together the results of all the previous sections to prove Theorem 1.3. The goal of Section 8 is to prove Theorem 1.4. We start with Lemmas 8.2 and 8.3 to establish the existence of certain decompositions in graphs of $\mathcal{C}$ , and then proceed with the sliding window argument. Section 9 is devoted to the proof of Theorem 9.1. The proof of Theorem 1.5 is completed in Section 10. Finally, Section 11 discusses algorithmic consequences of Theorem 1.4 and Theorem 1.5.

2. Special tree decompositions and connectivity

In this section we have collected several results from the literature that we need; for some of them we have also proved our own versions, tailored specifically to our needs. Along the way we also introduce some notation.

2.1. Connectivity

We start with a result on connectivity. For two vertices $u,v\in G$ and a set $X\subseteq G\setminus\{u,v\}$ we say that $X$ separates $u$ from $v$ if $P^{*}\cap X\neq\emptyset$ for every path $P$ of $G$ with ends $u$ and $v$ . The following is a well-known variant of a classical theorem due to Menger [30]:

Theorem 2.1 (Menger [30]).

Let $k\geq 1$ be an integer, let $G$ be a graph and let $u,v\in G$ be distinct and non-adjacent. Then either there exists a set $M\subseteq G\setminus\{u,v\}$ with $|M|<k$ such that $M$ separates $u$ and $v$ in $G$ , or $uv$ is a $k$ -banana in $G$ .

2.2. Lean tree decompositions

We adopt notation from [3]: For a tree $T$ and vertices $t,t^{\prime}\in V(T)$ , we denote by $tTt^{\prime}$ the unique path in $T$ from $t$ to $t^{\prime}$ . Let $(T,\chi)$ be a tree decomposition of a graph $G$ . For every $uv\in E(T)$ , the adhesion at $uv$ , denoted by $\operatorname{adh}(u,v)$ , is the set $\chi(u)\cap\chi(v)$ . For $u,v\in T$ such that $uv\not\in E(T)$ (in particular, if $u=v$ ), we set $\operatorname{adh}(u,v)=\emptyset$ . We define $\operatorname{adh}(T,\chi)=\max_{uv\in E(T)}|\operatorname{adh}(u,v)|$ . For every $x\in V(T)$ , the torso at $x$ , denoted by $\hat{\chi}(x)$ , is the graph obtained from the bag $\chi(x)$ by, for each $y\in N_{T}(x)$ , adding an edge between every two non-adjacent vertices $u,v\in\operatorname{adh}(x,y)$ .

In the proof of Theorem 1.3 and Theorem 1.5, we will use a special kind of a tree decomposition that we present next. Let $k>0$ be an integer. A tree decomposition $(T,\chi)$ is called $k$ -lean if the following hold:

•

$\operatorname{adh}(T,\chi)<k$ ; and
•

for all $t,t^{\prime}\in V(T)$ and sets $Z\subseteq\chi(t)$ and $Z^{\prime}\subseteq\chi(t^{\prime})$ with $|Z|=|Z^{\prime}|\leq k$ , either $G$ contains $|Z|$ disjoint paths from $Z$ to $Z^{\prime}$ , or some edge $ss^{\prime}$ of $tTt^{\prime}$ satisfies $|\operatorname{adh}(s,s^{\prime})|<|Z|$ .

For a tree $T$ and an edge $tt^{\prime}$ of $T$ , we denote by $T_{t\rightarrow t^{\prime}}$ the component of $T\setminus t$ containing $t^{\prime}$ . Let $G_{t\rightarrow t^{\prime}}=G[\bigcup_{v\in T_{t\rightarrow t^{\prime}}}\chi(v)]$ . A tree decomposition $(T,\chi)$ is tight if for every edge $tt^{\prime}\in E(T)$ there is a component $D$ of $G_{t\rightarrow t^{\prime}}\setminus\chi(t)$ such that $\chi(t)\cap\chi(t^{\prime})\subseteq N(D)$ (and therefore $\chi(t)\cap\chi(t^{\prime})=N(D)$ ).

Next, we need a definition from [9]. Given a tree decomposition $(T,\chi)$ of an $n$ -vertex graph $G$ , its fatness is the vector $(a_{n},\dots,a_{0})$ where $a_{i}$ denotes the number of bags of $T$ of size $i$ . A tree decomposition $(T,\chi)$ of $G$ is $k$ -atomic if $\operatorname{adh}(T,\chi)<k$ and the fatness of $(T,\chi)$ is lexicographically minimum among all tree decompositions of $G$ with adhesion less than $k$ .

It was observed in [13] that [9] contains a proof of the following:

Theorem 2.2 (Bellenbaum and Diestel [9], see Carmesin, Diestel, Hamann, Hundertmark [13], see also Weißauer [38]).

Every $k$ -atomic tree decomposition is $k$ -lean.

We also need:

Theorem 2.3 (Weißauer [38]).

Every $k$ -atomic tree decomposition is tight.

Using ideas similar to those of Weißauer [38] and using Theorems 2.1 and 2.2, we prove:

Theorem 2.4.

Let $G$ be a graph, let $k\geq 1$ and let $(T,\chi)$ be $3k$ -atomic tree decomposition of $G$ . Let $t_{0}\in V(T)$ and let $u,v\in G$ . Assume that $N[u]$ is not separated from $\chi(t_{0})\setminus u$ by a set of size less than $3k$ , and that $N[v]$ is not separated from $\chi(t_{0})\setminus v$ by a set of size less than $3k$ . Then $u$ is not separated from $v$ by a set of size less than $k$ , and consequently $uv$ is a $k$ -banana.

Proof.

By Theorems 2.2 and 2.3, we have that $(T,\chi)$ is tight and $3k$ -lean. Suppose that $u$ is separated from $v$ by a set of size less than $k$ . By Theorem 2.1, there exists a set $\mathcal{P}_{u}$ of pairwise vertex-disjoint (except $u$ ) paths, each with one end $u$ and the other end in $\chi(t_{0})\setminus u$ , and such that $|\mathcal{P}_{u}|=3k$ . Let $Z_{u}$ be the set of the ends of members of $\mathcal{P}_{u}$ in $\chi(t_{0})$ . Similarly, there exists a collection $\mathcal{P}_{v}$ of pairwise vertex-disjoint (except $v$ ) paths, each with one end $v$ and the other end in $\chi(t_{0})\setminus v$ , and such that $|\mathcal{P}_{v}|=3k$ . Let $Z_{v}$ be the set of the ends of members of $\mathcal{P}_{v}$ in $\chi(t_{0})$ . Since $(T,\chi)$ is $3k$ -lean, there is a collection $\mathcal{Q}$ of pairwise vertex-disjoint paths from $Z_{u}$ to $Z_{v}$ . Let $X$ be a set with $|X|<k$ separating $u$ from $v$ . Then, for every $u$ - $v$ path $P$ in $G$ we have $P^{*}\cap X\neq\emptyset$ . But since $|Z_{u}|=|Z_{v}|=|\mathcal{Q}|=3k$ , there is a path $Q\in\mathcal{Q}$ with ends $z_{u}\in Z_{u}$ and $z_{v}\in Z_{v}$ , a path $P_{u}\in\mathcal{P}_{u}$ from $u$ to $z_{u}$ , and a path $P_{v}\in\mathcal{P}_{v}$ from $v$ to $Z_{v}$ such that $X\cap(Q\cup(P_{u}\setminus u)\cup(P_{v}\setminus v))=\emptyset$ , contrary to the fact that $u\hbox{-}P_{u}\hbox{-}z_{u}\hbox{-}Q\hbox{-}z_{v}\hbox{-}P_{v}\hbox{-}v$ is a path from $u$ to $v$ in $G$ . This proves the first statement of the theorem. The second statement follows immediately by Theorem 2.1. ∎

We finish this subsection with a theorem about tight tree decompositions in theta-free graphs that was proved in [3]. Note that by Theorem 2.3, the following result applies in particular to $k$ -atomic tree decompositions for every $k$ .

A cutset $C\subseteq V(G)$ of $G$ is a (possibly empty) set of vertices such that $G\setminus C$ is disconnected. A clique cutset is a cutset that is a clique.

Theorem 2.5 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

Let $G$ be a theta-free graph and assume that $G$ does not admit a clique cutset. Let $(T,\chi)$ be a tight tree decomposition of $G$ . Then for every edge $t_{1}t_{2}$ of $T$ the graph $G_{t_{1}\rightarrow t_{2}}\setminus\chi(t_{1})$ is connected and $N(G_{t_{1}\rightarrow t_{2}}\setminus\chi(t_{1}))=\chi(t_{1})\cap\chi(t_{2})$ . Moreover, if $t_{0},t_{1},t_{2}\in V(T)$ and $t_{1},t_{2}\in N_{T}(t_{0})$ , then $\chi(t_{0})\cap\chi(t_{1})\neq\chi(t_{0})\cap\chi(t_{2})$ .

2.3. Catching a banana

In this subsection we discuss another important feature of tree decompositions that is needed in the proof of Theorem 1.3.

Theorem 2.6.

Let $G$ be a theta-free graph and let $a,b\in V(G)$ . Let $(T,\chi)$ be a tree decomposition of $G$ . Then there exist $t_{1},t_{2}\in V(T)$ (not necessarily distinct) such that for every path $P$ of $G$ with ends $a,b$ we have that $(\chi(t_{1})\cup\chi(t_{2}))\cap P^{*}\neq\emptyset.$ Moreover, if $D$ is a component of $G\setminus(\chi(t_{1})\cup\chi(t_{2}))$ , then $|N(D)\cap\{a,b\}|\leq 1$ .

Proof.

Let $\mathcal{P}$ be the set of all paths of $G$ with ends $a,b$ . For every $P\in\mathcal{P}$ let $T(P)=\{t\in T\ \;:\;\chi(t)\cap P^{*}\neq\emptyset\}$ . Since $P^{*}$ is a connected subgraph of $G$ , it follows that $T[T(P)]$ is a subtree of $T$ .

Let $H$ be a graph with vertex set $\mathcal{P}$ and such that $P_{1}P_{2}\in E(H)$ if and only if $T(P_{1})\cap T(P_{2})\neq\emptyset$ . Then, by the main result of [22], $H$ is a chordal graph.

(1) If $P_{1},P_{2}\in H$ are non-adjacent in $H$ , then $P_{1}^{*}$ , $P_{2}^{*}$ are disjoint and anticomplete to each other in $G$ .

Suppose that there exists $v\in P_{i}^{*}\cap P_{j}^{*}$ . Then for every $t\in T$ such that $v\in\chi(T)$ , we have $t\in T(P_{1})\cap T(P_{2})$ , contrary to the fact that $P_{1},P_{2}$ are non-adjacent in $H$ . Next suppose that there is an edge $v_{1}v_{2}$ of $G$ such that $v_{i}\in P_{i}^{*}$ . Let $t\in T$ be such that $v_{1},v_{2}\in\chi(t)$ . Then $t\in T(P_{1})\cap T(P_{2})$ , again contrary to the fact that $P_{1}$ is non-adjacent to $P_{2}$ in $H$ . This proves (2.3).

(2) $H$ has no stable set of size three.

Suppose $P_{1},P_{2},P_{3}$ is a stable set in $H$ . By (2.3) the sets $P_{1}^{*},P_{2}^{*},P_{3}^{*}$ are pairwise disjoint and anticomplete to each other in $G$ . But now $P_{1}\cup P_{2}\cup P_{3}$ is a theta with ends $a,b$ in $G$ , a contradiction. This proves (2.3).

Since $H$ is chordal, it follows from (2.3) and a result from [23] that there exist cliques $K_{1},K_{2}$ of $H$ such that $V(H)=K_{1}\cup K_{2}$ . Again by [23] we deduce that for each $i\in\{1,2\}$ , there exists $t_{i}\in K_{i}$ such that $t_{i}\in T(P)$ for every $P\in K_{i}$ . Consequently, $(\chi(t_{1})\cup\chi(t_{2}))\cap P^{*}\neq\emptyset$ for every $P\in\mathcal{P}$ , and the first statement of the theorem holds.

To prove the second statement, suppose $a,b\in N(D)$ for some component $D$ of $G\setminus(\chi(t_{1})\cup\chi(t_{2}))$ . Then $D\cup\{a,b\}$ is connected, and so there is a path $P$ from $a$ to $b$ with $P^{*}\subseteq D$ . But now $P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))=\emptyset$ , a contradiction. This completes the proof. ∎

2.4. Centers of tree decompositions.

We finish this section with a well-known theorem about tree decompositions. Recall that for a graph $G$ , a function $w:V(G)\rightarrow[0,1]$ is a normal weight function on $G$ if $w(V(G))=1$ , where we use the notation $w(X)$ to mean $\sum_{x\in X}w(x)$ for a set $X$ of vertices.

Let $G$ be a graph and let $(T,\chi)$ be a tree decomposition of $G$ . Let $w:V(G)\rightarrow[0,1]$ be a normal weight function on $G$ . A vertex $t_{0}$ of $T$ is a center of $(T,\chi)$ if for every $t^{\prime}\in N_{T}(t_{0})$ we have $w(G_{t_{0}\rightarrow t^{\prime}}\setminus\chi(t_{0}))\leq\frac{1}{2}$ .

The following is well-known; a proof can be found in [3], for example.

Theorem 2.7.

Let $(T,\chi)$ be a tree decomposition of a graph $G$ . Then $(T,\chi)$ has a center.

3. Wheels and star cutsets

Recall that a wheel in $G$ is a pair $(H,x)$ such that $H$ is a hole and $x$ is a vertex that has at least three neighbors in $H$ . Wheels play an important role in the study of even-hole-free and odd-signable graphs. Graphs in this classes that contain no wheels are much easier to handle; on the other hand, the presence of a wheel forces the graph to admit a certain kind of decomposition.

A sector of $(H,x)$ is a path $P$ of $H$ whose ends are distinct and adjacent to $x$ , and such that $x$ is anticomplete to $P^{*}$ . A sector $P$ is a long sector if $P^{*}$ is non-empty. We now define several types of wheels that we will need.

A wheel $(H,x)$ is a universal wheel if $x$ is complete to $H$ . A wheel $(H,x)$ is a twin wheel if $N(x)\cap H$ induces a path of length two. A wheel $(H,x)$ is a short pyramid if $|N(x)\cap H|=3$ and $x$ has exactly two adjacent neighbors in $H$ . A wheel is proper if it is not a twin wheel or a short pyramid. We say that $x\in V(G)$ is a hub if there exists $H$ such that $(H,x)$ is a proper wheel in $G$ . We denote by $\operatorname{Hub}(G)$ the set of all hubs of $G$ .

We need the following result, which was observed in [6]:

Theorem 3.1 (Abrishami, Chudnovsky, Vušković [6]).

Let $G\in\mathcal{C}$ and let $(H,v)$ be a proper wheel in $G$ . Then there is no component $D$ of $G\setminus N[v]$ such that $H\subseteq N[D]$ .

We will revisit this result in Section 8. A star cutset in a graph $G$ is a cutset $S\subseteq V(G)$ such that either $S=\emptyset$ or for some $x\in S$ , $S\subseteq N[x]$ . A large portion of this paper is devoted to dealing with hubs and star cutsets arising from them in graphs in $\mathcal{C}$ , but in the remainder of this section we focus on the case when $\operatorname{Hub}(G)$ is very restricted. To do that, we use a result from [3]:

Theorem 3.2 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

For every integer $t\geq 1$ there exists an integer $k=k(t)$ such that if $G\in\mathcal{C}_{t}$ and $xy$ is a $k$ -banana in $G$ , then $N(x)\cap\operatorname{Hub}(G)\neq\emptyset$ and $N(y)\cap\operatorname{Hub}(G)\neq\emptyset$ .

We immediately deduce:

Theorem 3.3.

For every integer $t\geq 1$ , there exists a constant $k=k(t)$ with the following property. Let $G\in\mathcal{C}_{t}$ and let $a,b\in V(G)$ be non-adjacent. Assume that $Hub(G)\subseteq\{a,b\}$ . Then $ab$ is not a $k$ -banana in $G$ . In particular, if $Hub(G)=\emptyset$ , there is no $k$ -banana in $G$ .

We will also use the following:

Theorem 3.4 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

For every integer $t\geq 1$ , there exists an integer $\gamma=\gamma(t)$ such that every $G\in\mathcal{C}_{t}$ with $\operatorname{Hub}(G)=\emptyset$ satisfies $\operatorname{tw}(G)\leq\gamma-1$ .

4. Stable sets of safe hubs

As we discussed in Section 1, in the course of the proof of Theorem 1.3, we will repeatedly decompose the graph by star cutsets arising from a stable set of appropriately chosen hubs (using Theorem 3.1). In this section, we prepare the tools for handling one such step: a stable set of safe hubs.

Throughout this section, we fix the following: Let $t,d\in\mathbb{N}$ and let $G\in\mathcal{C}_{t}$ be a graph with $|V(G)|=n$ . Let $m=k+2d$ where $k=k(t)$ is as in Theorem 3.2. Let $a,b\in V(G)$ such that $ab$ is a $(4m+2)(m-1)$ -banana in $G$ , and let $\mathcal{P}$ be the set of all paths in $G$ with ends $a,b$ . Let $(T,\chi)$ be an $m$ -atomic tree decomposition of $G$ . By Theorems 2.2 and 2.3, we have that $(T,\chi)$ is tight and $m$ -lean. By Theorem 2.6, there exist $t_{1},t_{2}\in T$ such that $P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))\neq\emptyset$ for every $P\in\mathcal{P}$ . We observe:

Lemma 4.1.

We have $a,b\in\chi(t_{1})\cup\chi(t_{2})$ .

Proof.

Suppose that $a\not\in\chi(t_{1})\cup\chi(t_{2})$ . Let $D$ be the component of $G\setminus(\chi(t_{1})\cup\chi(t_{2}))$ such that $a\in D$ . Then $N(D)$ is contained in the union of at most two adhesions of $(T,\chi)$ , at most one for each of $\chi(t_{1})$ and $\chi(t_{2})$ . Since $(T,\chi)$ is $m$ -lean, it follows that $|N(D)|\leq 2(m-1)$ . Since $P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))\neq\emptyset$ for every $P\in\mathcal{P}$ , it follows that $P^{*}\cap N(D)\neq\emptyset$ for every $P\in\mathcal{P}$ . But then $\mathcal{P}$ contains at most $|N(D)|$ pairwise internally vertex-disjoint paths, contrary to the fact that $ab$ is a $(4m+2)(m-1)$ -banana in $G$ . ∎

We say that a vertex $v$ is $d$ -safe if $|N(v)\cap\operatorname{Hub}(G)|\leq d$ . The goal of the next two lemmas is to classify $d$ -safe vertices into “good ones” and “bad ones,” and show that the bad ones are rare. Let $t_{0}\in V(T)$ . A vertex $v\in V(G)$ is $t_{0}$ -cooperative if either $v\not\in\chi(t_{0})$ , or $\deg_{\hat{\chi}(t_{0})}(v)<2m(m-1)$ . It was shown in [3] that $t_{0}$ -cooperative vertices have the following important property (note that [3] assumed that $t_{0}$ is a center of $(T,\chi)$ , but this was not used in the proof of the following lemma):

Lemma 4.2 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

Let $t_{0}\in V(T)$ . If $u,v\in\chi(t_{0})$ are $d$ -safe and not $t_{0}$ -cooperative, then $u$ is adjacent to $v$ .

A vertex $v\in G$ is $ab$ -cooperative if there exists a component $D$ of $G\setminus N[v]$ such that $a,b\in N[D]$ . We show:

Lemma 4.3.

If $v\in G$ is $d$ -safe and not $ab$ -cooperative, then $v$ is adjacent to both $a$ and $b$ . In particular, the set of vertices that are not $ab$ -cooperative is a clique.

Proof.

Suppose $v$ is non-adjacent to $a$ and $v$ is not $ab$ -cooperative. Since $ab$ is a $(4m+2)(m-1)$ -banana and $(4m+2)(m-1)\geq k+d$ , we can choose $P_{1},\dots,P_{k+d}\in\mathcal{P}$ such that the sets $P_{1}^{*},\dots,P_{k+d}^{*}$ are pairwise vertex-disjoint. Since $v$ is not $ab$ -cooperative, it follows that $v$ has a neighbors in each of $P_{1}^{*},\dots,P_{k+d}^{*}$ , and so there exist pairwise vertex-disjoint paths $Q_{1},\dots Q_{k+d}$ , each with ends $a,v$ , and such that $Q_{i}^{*}\subseteq P_{i}^{*}$ for every $i\in\{1,\dots,k+d\}$ . Since $v$ is $d$ -safe, we may assume (by renumbering the paths $Q_{1},\dots,Q_{k+d}$ if necessary), that $N(v)\cap\operatorname{Hub}(G)\cap Q_{i}=\emptyset$ for $i\in\{1,\dots,k\}$ . But now we get a contradiction applying Theorem 3.2 to the graph $H=Q_{1}\cup\dots\cup Q_{k}$ and the $k$ -banana $av$ in $H$ . This proves the first statement of 4.3. Since $G$ is $C_{4}$ -free, the second statement follows. ∎

In view of Theorem 2.6, it would be convenient if we could work with the graph $\hat{\chi}(t_{1})\cup\hat{\chi}(t_{2})$ , but unfortunately this graph may not be in $\mathcal{C}_{t}$ , or even $\mathcal{C}$ . In [3], a tool was designed to construct a safe alternative to torsos by adding a set of vertices for each adhesion, rather than turning the adhesion into a clique:

Theorem 4.4 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

Let $t_{0}\in T$ . Assume that $G$ does not admit a clique cutset. For every $t\in N_{T}(t_{0})$ , there exists $\operatorname{Conn}(t_{0},t)\subseteq G_{t_{0}\rightarrow t}$ such that

(1)

$\operatorname{adh}(t,t_{0})=\chi(t)\cap\chi(t_{0})\subseteq\operatorname{Conn}(t_{0},t)$ .
(2)

$\operatorname{Conn}(t_{0},t)\setminus\chi(t_{0})$ is connected and $N(\operatorname{Conn}(t_{0},t)\setminus\chi(t_{0}))=\chi(t_{0})\cap\chi(t)$ .
(3)

No vertex of $\operatorname{Conn}(t_{0},t)\setminus\chi(t_{0})$ is a hub in the graph $(G\setminus G_{t_{0}\rightarrow t})\cup\operatorname{Conn}(t_{0},t)$ .

Similarly to the construction in [3], we now define a graph that is a safe alternative to $\hat{\chi}(t_{1})\cup\hat{\chi}(t_{2})$ . Let $S^{\prime}$ be a stable set of hubs of $G$ with $S^{\prime}\cap\{a,b\}=\emptyset$ , and assume that every $s\in S^{\prime}$ is $d$ -safe. Let $S_{bad}$ denote the set of all vertices in $S^{\prime}$ that are common neighbors of $a$ and $b$ , or are not $t_{1}$ -cooperative, or are not $t_{2}$ -cooperative. Since $S^{\prime}$ is stable, Lemma 4.2 implies that $|S_{bad}|\leq 3$ . By Lemma 4.3, every vertex in $S^{\prime}\setminus S_{bad}$ is $ab$ -cooperative. If $t_{1}\neq t_{2}$ , for $i\in\{1,2\}$ , let $t_{i}^{\prime}$ be the neighbor of $t_{i}$ in the (unique) path in $T$ from $t_{1}$ to $t_{2}$ . If $t_{1}=t_{2}$ , set $t_{1}^{\prime}=t_{2}^{\prime}=t_{1}$ . Let $S=S^{\prime}\setminus S_{bad}$ and set

\beta(S^{\prime})=\left(\chi(t_{1})\cup\chi(t_{2})\cup\left(\bigcup_{i\in\{1,2\}}\bigcup_{t\in N_{T}(t_{i})\setminus\{t_{i}^{\prime}\}}\operatorname{Conn}(t_{i},t)\right)\right)\setminus S_{bad}.

Write $\beta=\beta(S^{\prime})$ . We fix $S^{\prime},S,S_{bad}$ and $\beta$ throughout this section. It follows that for every $i\in\{1,2\}$ and for every $t\in N_{T}(t_{i})\setminus\{t_{i}^{\prime}\}$ , we have that $\beta\subseteq(G\setminus G_{t_{i}\rightarrow t})\cup\operatorname{Conn}(t_{i},t)$ . Let $X\subseteq\beta$ . For $i\in\{1,2\}$ , we define $\delta_{i}(X)$ to be the set of all vertices $t\in N_{T}(t_{i})\setminus\{t_{i}^{\prime}\}$ such that $X\cap(G_{t_{i}\rightarrow t}\setminus(\chi(t_{1})\cup\chi(t_{2})))\neq\emptyset$ ; let $\delta(X)=\delta_{1}(X)\cup\delta_{2}(X)$ . Write

\Delta(X)=\bigcup_{t\in\delta_{1}(X)}\operatorname{adh}(t_{1},t)\cup\bigcup_{t\in\delta_{2}(X)}\operatorname{adh}(t_{2},t).

Next we summarize several key properties of $\beta$ .

Lemma 4.5.

Suppose that $G$ does not admit a clique cutset and let $s\in S\cap\operatorname{Hub}(\beta)$ . Then the following hold.

(1)

$s\in\chi(t_{1})\cup\chi(t_{2})$ .
(2)

For $i\in\{1,2\}$ , we have $|N_{\hat{\chi}(t_{i})}(s)|<2m(m-1)$ .
(3)

$|N_{\chi(t_{1})\cup\chi(t_{2})}(s)|<4m(m-1)$ .
(4)

There is a component $B(s)$ of $\beta\setminus N[s]$ such that $a,b\in N[B(s)]$ .

Proof.

Let $s\in S\cap\operatorname{Hub}(\beta)$ . It follows from Theorem 4.4(3) that $s\in\chi(t_{1})\cup\chi(t_{2})$ . Since $s$ is $t_{i}$ -cooperative for $i\in\{1,2\}$ , we have that $|N_{\hat{\chi}(t_{i})}(s)|<2m(m-1)$ , and the second assertion of the theorem holds. Since $\chi(t_{i})$ is a subgraph of $\hat{\chi}(t_{i})$ , the third assertion follows immediately from the second.

We now prove the fourth assertion. Suppose there is no component $D$ of $\beta\setminus N[s]$ such that $a,b\in N[D]$ . Write

\hat{\Delta}(s)=N_{\chi(t_{1})\cup\chi(t_{2})}[s]\cup\operatorname{adh}(t_{1},t_{1}^{\prime})\cup\operatorname{adh}(t_{2},t_{2}^{\prime})\cup\Delta(N_{\beta}[s]).

Since $\hat{\Delta}(s)\subseteq N_{\hat{\chi}(t_{1})}[s]\cup N_{\hat{\chi}(t_{2})}[s]\cup\operatorname{adh}(t_{1},t_{1}^{\prime})\cup\operatorname{adh}(t_{2},t_{2}^{\prime})$ , and $\operatorname{adh}(T,\chi)\leq m-1$ , and by the second assertion of the theorem, we have that $|\hat{\Delta}(s)|<(4m+2)(m-1)$ .

(3) $\hat{\Delta}(s)\cap Q^{*}\neq\emptyset$ for every path in $\beta$ with ends $a,b$ .

Suppose there is a path $Q$ in $\beta$ with ends $a,b$ such that $\hat{\Delta}(s)\cap Q^{*}=\emptyset$ . Since there is no component $D$ of $\beta\setminus N[s]$ such that $a,b\in N[D]$ , it follows that $N_{\beta}(s)\cap Q^{*}\neq\emptyset$ . Consequently, there is an $i\in\{1,2\}$ and $t\in N_{T}(t_{i})\setminus\{t_{i}^{\prime}\}$ such that $N_{\beta}(s)\cap Q^{*}\cap\operatorname{Conn}(t_{i},t)\neq\emptyset$ . Since $Q$ is a path from $a$ to $b$ in $G$ , it follows that $Q^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))\neq\emptyset$ . Since $Q^{*}$ is connected and $\operatorname{adh}(t_{i},t)$ separates $G_{t_{i}\rightarrow t}\setminus\chi(t_{i})$ from $(\chi(t_{1})\cup\chi(t_{2}))\setminus\chi(t)$ , we deduce that $Q^{*}\cap\operatorname{adh}(t_{i},t)\neq\emptyset$ . But since $s$ has a neighbor in $\operatorname{Conn}(t_{i},t)$ , it follows that $\operatorname{adh}(t_{i},t)\subseteq\hat{\Delta}(s)$ , a contradiction. This proves (4).

(4) $P^{*}\cap\hat{\Delta}(s)\neq\emptyset$ for every path $P$ of $G$ with ends $a,b$ .

Let $P$ be a path in $G$ with ends $a,b$ . We prove by induction on $|P\setminus\beta|$ that $\hat{\Delta}(s)\cap P^{*}\neq\emptyset$ . If $P\subseteq\beta$ , the claim follows from (4), and this is the base case.

Let $p\in P^{*}\setminus\beta$ and let $t_{0}\in T$ such that $p\in\chi(t_{0})$ . Suppose first that $t_{0}$ belongs to the component $T^{\prime}$ of $T\setminus\{t_{1},t_{2}\}$ such that $t_{1}^{\prime}\in T^{\prime}$ . Then $t_{2}^{\prime}\in T^{\prime}$ . Since $P^{*}$ is connected and $P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))\neq\emptyset$ , it follows that $P^{*}\cap(\operatorname{adh}(t_{1},t_{1}^{\prime})\cup\operatorname{adh}(t_{2},t_{2}^{\prime}))\neq\emptyset$ , and so $P^{*}\cap\hat{\Delta}(s)\neq\emptyset$ . Thus we may assume that $p\in G_{t_{1}\rightarrow t}\setminus\chi(t_{1})$ for some $t\in N_{T}(t_{1})\setminus\{t_{1}^{\prime}\}$ . Since $P^{*}$ is connected and $P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))\neq\emptyset$ , it follows that $P^{*}\cap\operatorname{adh}(t_{1},t)\neq\emptyset$ . Write $P=p_{1}\hbox{-}\cdots\hbox{-}p_{l}$ where $p_{1}=a$ and $p_{l}=b$ . Let $i$ be minimum and $j$ maximum such that $p_{i}\in\operatorname{adh}(t_{1},t)$ . Since $p\in G_{t_{1}\rightarrow t}\setminus\chi(t_{1})$ , it follows that $i<j-1$ . By Theorem 4.4(2) there is a path $R$ form $p_{i}$ to $p_{j}$ with $R^{*}\subseteq\operatorname{Conn}(t_{1},t)$ . Then $P^{\prime}=p_{1}\hbox{-}\cdots\hbox{-}p_{i}\hbox{-}R\hbox{-}p_{j}\hbox{-}\dots\hbox{-}p_{l}$ is a path from $a$ to $b$ in $G$ with $|P^{\prime}\setminus\beta|<|P\setminus\beta|$ . Inductively, $\hat{\Delta}(s)\cap P^{\prime*}\neq\emptyset$ . But $\hat{\Delta}(s)\subseteq\chi(t_{1})\cup\chi(t_{2})$ and $P^{\prime*}\cap(\chi(t_{1})\cup\chi(t_{2}))\subseteq P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))$ , and so $\hat{\Delta}(s)\cap P^{*}\neq\emptyset$ . This proves (4).

Since $ab$ is a $(4m+2)(m-1)$ -banana in $G$ , and since $\hat{\Delta}(s)<(4m+2)(m-1)$ , (4) leads to a contradiction. This completes the proof.

∎

Recall that a separation of $\beta$ is a triple $(X,Y,Z)$ of pairwise disjoint subsets of $\beta$ with $X\cup Y\cup Z=\beta$ such that $X$ is anticomplete to $Z$ . We are now ready to move on to star cutsets. We will work in the graph $\beta$ and take advantage of its special properties. At the end of the section we will explain how to connect our results back to the graph $G$ that we are interested in.

As in other papers in the series, we associate a certain unique star separation to every vertex of $S\cap\operatorname{Hub}(\beta)$ . The separation here is chosen differently from the way it was done in the past, but the behavior we observe is similar. The reason for the difference is that unlike in [2] or [6], our here goal is to disconnect two given vertices, rather than find a “balanced separator” in the graph (more on this in Section 10).

Let $v\in S\cap\operatorname{Hub}(\beta)$ . By Theorem 4.5, there is a component $D$ of $\beta\setminus N[v]$ such that $a,b\in N[D]$ . Since $s\not\in S_{bad}$ , it follows that $s$ is not complete to $\{a,b\}$ ; consequently $D\cap\{a,b\}\neq\emptyset$ , and so $D$ is unique. Let $B(v)$ be the unique component of $\beta\setminus N[v]$ such that $a,b\in N[B(v)]$ (this choice of $B(v)$ is different from what we have done in earlier papers). Let $C(v)=N(B(v))\cup\{v\}$ , and finally, let $A(v)=\beta\setminus(B(v)\cup C(v))$ . Then $(A(v),C(v),B(v))$ is the canonical star separation of $\beta$ corresponding to $v$ . The next lemma is a key step in the proof of the fact that decomposing by canonical star separations makes the graph simpler. We show:

Lemma 4.6.

The vertex $v$ is not a hub of $\beta\setminus A(v)$ .

Proof.

Suppose that $(H,v)$ is a proper wheel in $\beta\setminus A(v)$ . Then $H\subseteq N[B(v)]$ , contrary to Theorem 3.1. This proves Lemma 4.6. ∎

We need just a little more set up. Let $\mathcal{O}$ be a linear order on $S\cap\operatorname{Hub}(\beta)$ . Following [4], we say that two vertices of $S\cap\operatorname{Hub}(\beta)$ are star twins if $B(u)=B(v)$ , $C(u)\setminus\{u\}=C(v)\setminus\{v\}$ , and $A(u)\cup\{u\}=A(v)\cup\{v\}$ . (Note that every two of these conditions imply the third.)

Let $\leq_{A}$ be a relation on $S\cap\operatorname{Hub}(\beta)$ defined as follows:

\hskip 65.44142ptx\leq_{A}y\ \ \ \text{ if}\ \ \ \begin{cases}x=y;\\ \text{$x$ and $y$ are star twins and $\mathcal{O}(x)<\mathcal{O}(y)$; or}\\ \text{$x$ and $y$ are not star twins and }y\in A(x).\\ \end{cases}

Note that if $x\leq_{A}y$ , then either $x=y$ , or $y\in A(x).$

The conclusion of the next lemma is the same as of Lemma 6.2 from [5], but the assumptions are different.

Lemma 4.7.

If $y\in A(x)$ , then $A(y)\cup\{y\}\subseteq A(x)\cup\{x\}$ .

Proof.

Since $C(y)\subseteq N[y]$ and $y$ is anticomplete to $B(x)$ , we have $B(x)\subseteq G\setminus N[y]$ . Since $B(x)$ is connected, there is a component $D$ of $G\setminus N[y]$ such that $B(x)\subseteq D$ . Since $x\not\in S_{bad}$ , it follows that $\{a,b\}\cap B(x)\neq\emptyset$ , and so $D=B(y)$ . Let $v\in C(x)\setminus\{x\}$ . Then $v$ has a neighbor in $B(x)$ and thus in $B(y)$ . If $v\in N[y]$ , then $v\in C(y)$ . If $v\not\in N[y]$ , then $v\in B(y)$ . It follows that $C(x)\setminus\{x\}\subseteq C(y)\cup B(y)$ . But now $A(y)\setminus\{x\}\subseteq A(x)$ , as required. This proves Lemma 4.7. ∎

The proofs of the next two lemmas are identical to the proofs of Lemmas 6.3 and Lemma 6.4 in [5] (using Lemma 4.7 instead of Lemma 6.2 of [5]), and we omit them.

Lemma 4.8.

$\leq_{A}$ is a partial order on $S\cup\operatorname{Hub}(\beta)$ .

In view of Lemma 4.8, let $\operatorname{Core}(S^{\prime})$ be the set of all $\leq_{A}$ -minimal elements of $S\cap\operatorname{Hub}(\beta)$ .

Lemma 4.9.

Let $u,v\in\operatorname{Core}(S^{\prime})$ . Then $A(u)\cap C(v)=C(u)\cap A(v)=\emptyset$ .

We have finally reached our goal: we can define a subgraph of $G$ that is simpler than $G$ itself, but carries all the information we need. Define

\beta^{A}(S^{\prime})=\bigcap_{v\in\operatorname{Core}(S^{\prime})}(B(v)\cup C(v)).

The next theorem summarizes the important aspects of the behavior of $\beta^{A}(S^{\prime})$ .

Theorem 4.10.

The following hold:

(1)

For every $v\in\operatorname{Core}(S^{\prime})$ , we have $C(v)\subseteq\beta^{A}(S^{\prime})$ .
(2)

For every $v\in\operatorname{Core}(S^{\prime})$ , $|C(v)\cap(\chi(t_{1})\cup\chi(t_{2}))|\leq 4m(m-1)$ .
(3)

For every $v\in\operatorname{Core}(S^{\prime})$ , $|\Delta(C(v))|\leq 4m(m-1)$ .
(4)

For every component $D$ of $\beta\setminus\beta^{A}(S^{\prime})$ , there exists $v\in\operatorname{Core}(S^{\prime})$ such that $D\subseteq A(v)$ . Further, if $D$ is a component of $\beta\setminus\beta^{A}(S^{\prime})$ and $v\in\operatorname{Core}(S^{\prime})$ such that $D\subseteq A(v)$ , then $N_{\beta}(D)\subseteq C(v)$ .
(5)

$S^{\prime}\cap\operatorname{Hub}(\beta^{A}(S^{\prime}))=\emptyset$ .

Proof.

(1) is immediate from Lemma 4.9, and (2) follows from Lemma 4.5.

Next we prove (3). By Lemma 4.5(1), it follows that $v\in\chi(t_{1})\cup\chi(t_{2})$ . Observe that if $t\in T\setminus\{t_{1},t_{2},t_{1}^{\prime},t_{2}^{\prime}\}$ and $(G_{t_{i}\rightarrow t}\setminus(\chi(t_{1})\cup\chi(t_{2}))\cap C(v)\neq\emptyset$ for some $i\in\{1,2\}$ , then $v\in\chi(t_{1})\cap\chi(t)$ or $v\in\chi(t_{2})\cap\chi(t)$ . It follows that for $i\in\{1,2\}$ , $\operatorname{adh}(t_{i},t)\subseteq\Delta(C(v))$ only if $v\in\operatorname{adh}(t_{i},t)$ . Consequently, $|\Delta(C(v))|\leq\deg_{\hat{\chi}(t_{1})}(v)+\deg_{\hat{\chi}(t_{2})}(v)$ , and so (3) follows from Lemma 4.5(3).

Next we prove (4). Let $D$ be a component of $\beta\setminus\beta^{A}(S^{\prime})$ . Since $\beta\setminus\beta^{A}(S^{\prime})=\bigcup_{v\in\operatorname{Core}(S^{\prime})}A(v)$ , there exists $v\in\operatorname{Core}(S^{\prime})$ such that $D\cap A(v)\neq\emptyset$ . If $D\setminus A(v)\neq\emptyset$ , then, since $D$ is connected, it follows that $D\cap N(A(v))\neq\emptyset$ ; but then $D\cap C(v)\neq\emptyset$ , contrary to (1). Since $N_{\beta}(D)\subseteq\beta^{A}(S^{\prime})$ and $N_{\beta}(D)\subseteq A(v)\cup C(v)$ , it follows that $N_{\beta}(D)\subseteq C(v)$ . This proves (4).

To prove (5), let $u\in S^{\prime}\cap\operatorname{Hub}(\beta^{A}(S^{\prime}))$ . Since $\beta^{A}(S^{\prime})\subseteq\beta$ , we deduce that $u\not\in S_{bad}$ , and so $u\in S\cap\operatorname{Hub}(\beta)$ . By Theorem 4.5(1), we have that $u\in\chi(t_{1})\cup\chi(t_{2})$ . By Lemma 4.6, it follows that $\beta^{A}(S^{\prime})\not\subseteq B(u)\cup C(u)$ , and therefore $u\not\in\operatorname{Core}(S^{\prime})$ . But then $u\in A(v)$ for some $v\in\operatorname{Core}(S^{\prime})$ , and so $u\not\in\beta^{A}(S^{\prime})$ , a contradiction. This proves (5) and completes the proof of Theorem 4.10. ∎

We now explain how $\beta^{A}(S^{\prime})$ is used. In the course of the proof of Theorem 1.3, we will inductively obtain a small cutset separating $a$ from $b$ in $\beta^{A}(S^{\prime})$ . The next theorem allows us to transform this cutset into a cutset separating $a$ from $b$ in $\beta$ .

Theorem 4.11.

Let $(X,Y,Z)$ be a separation of $\beta^{A}(S^{\prime})$ such that $a\in X$ and $b\in Z$ . Let $Y^{\prime}=(Y\cup\bigcup_{s\in Y\cap\operatorname{Core}(S^{\prime})}C(s))\setminus\{a,b\}$ . Then

(1)

$Y^{\prime}$ separates $a$ from $b$ in $\beta$ .
(2)

$|Y^{\prime}\cap(\chi(t_{1})\cup\chi(t_{2}))|\leq|Y|+4m(m-1)|Y\cap\operatorname{Core}(S^{\prime})|$ .
(3)

$|\Delta(Y^{\prime})\setminus\Delta(Y)|\leq 4m(m-1)|Y\cap\operatorname{Core}(S^{\prime})|$ .

Proof.

Suppose that $Y^{\prime}$ does not separate $a$ from $b$ in $\beta$ . Let $P$ be a path from $a$ to $b$ in $\beta\setminus Y^{\prime}$ . Let $D_{a},D_{b}$ be the components of $\beta^{A}(S^{\prime})\setminus Y$ such that $a\in D_{a}$ and $b\in D_{b}$ . Since the first vertex of $P$ is in $D_{a}$ and the last vertex of $P$ is in $D_{b}$ , it follows that there is a subpath $Q$ of $P$ such that $Q$ has ends $u,v$ and:

•

The vertex $u$ is in $D_{a}$ ;
•

$Q^{*}\cap\beta^{A}(S^{\prime})=\emptyset$ ;
•

The vertex $v$ is in $\beta^{A}(S^{\prime})\setminus D_{a}$ .

See Figure 2. Since $Y\subseteq Y^{\prime}$ , it follows that $v$ is in a component $D_{c}$ of $\beta^{A}(S^{\prime})\setminus Y$ with $D_{c}\neq D_{a}$ (possibly $D_{c}=D_{b}$ ). This implies that $uv\not\in E(G)$ , and so $Q^{*}\neq\emptyset$ .

By Theorem 4.10(4), there is an $s\in\operatorname{Core}(S^{\prime})\subseteq\beta^{A}(S^{\prime})$ such that $Q^{*}\subseteq A(s)$ and $u,v\in C(s)$ . Since $D_{a},D_{c}$ are distinct components of $\beta^{A}(S^{\prime})\setminus Y$ , and since $N_{\beta^{A}(S^{\prime})}(s)$ meets both $D_{a}$ and $D_{b}$ , it follows that $s\in Y$ . Therefore $C(s)\setminus\{a,b\}\subseteq Y^{\prime}$ , and so $u,v\in\{a,b\}$ . Since $u\neq v$ , it follows that $u=a$ and $b=v$ , and so $a,b\in N(s)$ , contrary to the fact that $s\not\in S_{bad}$ . This proves the first assertion of the theorem.

The second assertion follows from Theorem 4.10(2), and the third assertion follows from Theorem 4.10(3). ∎

We finish this section with a theorem that allows us to transform a cutset separating $a$ from $b$ in $\beta$ into a cutset separating $a$ from $b$ in $G$ .

Theorem 4.12.

Assume that $G$ does not admit a clique cutset. Let $(X,Y,Z)$ be a separation of $\beta$ such that $a\in X$ and $b\in Z$ . Let

Y^{\prime}=(S_{bad}\cup\operatorname{adh}(t_{1},t_{1}^{\prime})\cup\operatorname{adh}(t_{2},t_{2}^{\prime})\cup(Y\cap(\chi(t_{1})\cup\chi(t_{2})))\cup\Delta(Y))\setminus\{a,b\}.

Then $Y^{\prime}$ separates $a$ from $b$ in $G$ .

Proof.

Suppose not. Let $D_{a},D_{b}$ be the components of $\beta\setminus Y$ such that $a\in D_{a}$ and $b\in D_{b}$ . Since $Y^{\prime}$ does not separate $a$ from $b$ in $G$ , and $Y^{\prime}\cap\{a,b\}=\emptyset$ , there is a component $D$ of $G\setminus Y^{\prime}$ such that $a,b\in D$ . Let $P$ be a path from $a$ to $b$ with $P\subseteq D$ .

As before, it follows that there is a subpath $Q$ of $P$ such that $Q$ has ends $u,v$ and:

•

The vertex $u$ is in $D_{a}$ ;
•

$Q^{*}\cap\beta=\emptyset$ ;
•

The vertex $v$ is in $\beta\setminus D_{a}$ .

We first consider the case that $v\in Y$ (and so $v\in P^{*}$ ). Then $v\in\operatorname{Conn}(t_{i},t)\setminus(\chi(t_{1})\cup\chi(t_{2}))$ for some $i\in\{1,2\}$ and $t\in N_{T}(t_{i})\setminus\{t_{i}^{\prime}\}$ . It follows that $t\in\delta_{i}(Y)$ , and so $\operatorname{adh}(t_{i},t)\setminus\{a,b\}\subseteq\Delta(Y)\setminus\{a,b\}\subseteq Y^{\prime}$ . It follows that $P^{*}\cap\operatorname{adh}(t_{i},t)=\emptyset$ , and from Theorem 2.6, we know that $P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))\neq\emptyset$ . Therefore, $P^{*}\cap(G_{t_{i}\rightarrow t}\setminus(\chi(t_{1})\cup\chi(t_{2})))=\emptyset$ , contrary to the fact that $v\in\operatorname{Conn}(t_{i},t)\setminus(\chi(t_{1})\cup\chi(t_{2}))$ . This is a contradiction, and proves that $v\not\in Y$ .

Therefore, there is a component $D_{c}\neq D_{a}$ (possibly $D_{b}=D_{c}$ ) of $\beta\setminus Y$ such that $v\in D_{c}$ . Since there are no edges between $D_{a}$ and $D_{c}$ , it follows that $Q^{*}\neq\emptyset$ .

Consequently, there is a component $\tilde{T}$ of $T\setminus\{t_{1},t_{2}\}$ such that $Q^{*}\subseteq(\bigcup_{t\in\tilde{T}}\chi(t))\setminus(\chi(t_{1})\cup\chi(t_{2}))$ . Let $\tilde{D}$ be the component of $G\setminus(\chi(t_{1})\cup\chi(t_{2}))$ such that $Q^{*}\subseteq\tilde{D}$ . By Theorem 2.6, it follows that $|N(\tilde{D})\cap\{a,b\}|\leq 1$ .

Suppose first that $t_{1}^{\prime}\in\tilde{T}$ . Then $t_{1}^{\prime}\neq t_{2}$ , and $t_{2}^{\prime}\neq t_{1}$ , and $t_{2}^{\prime}\in\tilde{T}$ . Since $N_{G}(\tilde{D})\subseteq\operatorname{adh}(t_{1},t_{1}^{\prime})\cup\operatorname{adh}(t_{2},t_{2}^{\prime})$ , it follows that $N_{G\setminus Y^{\prime}}(\tilde{D})\subseteq\{a,b\}$ , and so $|N_{G\setminus Y^{\prime}}(\tilde{D})|\leq 1$ . Since $P$ is a path with ends $a,b\not\in\tilde{D}$ and with $P\subseteq G\setminus Y^{\prime}$ , it follows that $P\cap\tilde{D}=\emptyset$ , and so $Q\cap\tilde{D}=\emptyset$ . Similarly, $t_{2}^{\prime}\not\in\tilde{T}$ .

We deduce that exactly one of $t_{1},t_{2}$ has a neighbor in $\tilde{T}$ (unless $t_{1}=t_{2}$ ), and this neighbor is unique; denote it by $t$ . By symmetry, we may assume that $tt_{1}\in E(T)$ , and so $\tilde{T}=T_{t_{1}\rightarrow t}$ . By Theorem 2.5, we deduce that $\tilde{D}=G_{t_{1}\rightarrow t}\setminus\chi(t_{1})$ . Since $u\in D_{a}\cap N(Q^{*})\subseteq\beta\cap N[\tilde{D}]$ , it follows that $u\in\operatorname{Conn}(t_{1},t)$ . Similarly, we have $v\in\operatorname{Conn}(t_{1},t)$ . Since $\operatorname{Conn}(t_{1},t)\setminus\chi(t_{1})$ is connected and $N(\operatorname{Conn}(t_{1},t)\setminus\chi(t_{1}))=\chi(t_{1})\cap\chi(t)$ by Theorem 4.4(2), it follows that $\operatorname{Conn}(t_{1},t)$ contains a path $Q^{\prime}$ with ends $u,v$ and interior in $\operatorname{Conn}(t_{1},t)\setminus\chi(t_{1})$ . Since $Q^{\prime}$ is a path from $D_{a}$ to $D_{c}$ in $\beta$ , it follows that $Y\cap Q^{\prime*}\neq\emptyset$ , and so $Y\cap(\operatorname{Conn}(t_{1},t)\setminus\chi(t_{1}))\neq\emptyset$ . Therefore, $Y\cap(G_{t_{1}\rightarrow t}\setminus\chi(t_{1}))\neq\emptyset$ , which implies that $\operatorname{adh}(t_{1},t)\setminus\{a,b\}\subseteq Y^{\prime}$ . It follows that $P^{*}\cap\operatorname{adh}(t_{1},t)=\emptyset$ , and from Theorem 2.6, we know that $P^{*}\cap(\chi(t_{1})\cup\chi(t_{2}))\neq\emptyset$ . Therefore, $P^{*}\cap(G_{t_{1}\rightarrow t}\setminus\chi(t_{1}))=\emptyset$ , contrary to the fact that $\emptyset\neq Q^{*}\subseteq G_{t_{1}\rightarrow t}\setminus\chi(t_{1})$ . This is a contradiction, and concludes the proof. ∎

5. Connectifiers

We start this section by describing minimal connected subgraphs containing the neighbors of a large number of vertices from a given stable set. We then use this result to deduce what a pair of two such subgraphs can look like (for the same stable set) assuming that they are anticomplete to each other and the graph in question is in $\mathcal{C}$ . This generalizes results from [2] and [3].

What follows is mostly terminology from [3], but there are also some new notions. Let $G$ be a graph, let $P=p_{1}\hbox{-}\cdots\hbox{-}p_{n}$ be a path in $G$ and let $X=\{x_{1},\dots,x_{k}\}\subseteq G\setminus P$ . We say that $(P,X)$ is an alignment if $N_{P}(x_{1})=\{p_{1}\}$ , $N_{P}(x_{k})=\{p_{n}\}$ , every vertex of $X$ has a neighbor in $P$ , and there exist $1<j_{2}<\dots<j_{k-1}<j_{k}=n$ such that $N_{P}(x_{i})\subseteq p_{j_{i}}\hbox{-}P\hbox{-}p_{j_{i+1}-1}$ for $i\in\{2,\dots,k-1\}$ . We also say that $x_{1},\dots,x_{k}$ is the order on $X$ given by the alignment $(P,X)$ . An alignment $(P,X)$ is wide if each of $x_{2},\dots,x_{k-1}$ has at least two non-adjacent neighbors in $P$ , spiky if each of $x_{2},\dots,x_{k-1}$ has a unique neighbor in $P$ and triangular if each of $x_{2},\dots,x_{k-1}$ has exactly two neighbors in $P$ and they are adjacent. An alignment is consistent if it is wide, spiky or triangular.

By a caterpillar we mean a tree $C$ with maximum degree three such that there exists a path $P$ of $C$ such that all vertices of degree $3$ in $C$ belong to $P$ . We call a minimal such path $P$ the spine of $C$ . By a subdivided star we mean a graph isomorphic to a subdivision of the complete bipartite graph $K_{1,\delta}$ for some $\delta\geq 3$ . In other words, a subdivided star is a tree with exactly one vertex of degree at least three, which we call its root. For a graph $H$ , a vertex $v$ of $H$ is said to be simplicial if $N_{H}(v)$ is a clique. We denote by $\mathcal{Z}(H)$ the set of all simplicial vertices of $H$ . Note that for every tree $T$ , $\mathcal{Z}(T)$ is the set of all leaves of $T$ . An edge $e$ of a tree $T$ is said to be a leaf-edge of $T$ if $e$ is incident with a leaf of $T$ . It follows that if $H$ is the line graph of a tree $T$ , then $\mathcal{Z}(H)$ is the set of all vertices in $H$ corresponding to leaf-edges of $T$ .

Let $H$ be a graph that is either a path, or a caterpillar, or the line graph of a caterpillar, or a subdivided star with root $r$ , or the line graph of a subdivided star with root $r$ . We define an induced subgraph of $H$ , denoted by $P(H)$ , which we will use throughout the paper. If $H$ is a path, we let $P(H)=H$ . If $H$ is a caterpillar, we let $P(H)$ be the spine of $H$ . If $H$ is the line graph of a caterpillar $C$ , let $P(H)$ be the path in $H$ consisting of the vertices of $H$ that correspond to the edges of the spine of $C$ . If $H$ is a subdivided star with root $r$ , let $P(H)=\{r\}$ . It $H$ is the line graph of a subdivided star $S$ with root $r$ , let $P(H)$ be the clique of $H$ consisting of the vertices of $H$ that correspond to the edges of $S$ incident with $r$ . The legs of $H$ are the components of $H\setminus P(H)$ .

Next, let $H$ be a caterpillar or the line graph of a caterpillar and let $S$ be a set of vertices disjoint from $H$ such that every vertex of $S$ has a unique neighbor in $H$ and $H\cap N(S)=\mathcal{Z}(H)$ . Let $X$ be the set of vertices of $H\setminus P(H)$ that have neighbors in $P(H)$ . Then the neighbors of elements of $X$ appear in $P(H)$ in order (there may be ties at the ends of $P(H)$ , which we break arbitrarily); let $x_{1},\dots,x_{k}$ be the corresponding order on $X$ . Now, order the vertices of $S$ as $s_{1},\dots,s_{k}$ where $s_{i}$ has a neighbor in the leg of $H$ containing $x_{i}$ for $i\in\{1,\dots,k\}$ . We say that $s_{1},\dots,s_{k}$ is the order on $S$ given by $(H,S)$ .

Next, let $H$ be an induced subgraph of $G$ that is a caterpillar, or the line graph of a caterpillar, or a subdivided star or the line graph of a subdivided star and let $X\subseteq G\setminus H$ be such that every vertex of $X$ has a unique neighbor in $H$ and $H\cap N(X)=\mathcal{Z}(H)$ . We say that $(H,X)$ is a consistent connectifier and it is spiky, triangular, stellar, or clique respectively. If $H$ is a single vertex and $X\subseteq N(H)$ , we also call $(H,X)$ a stellar connectifer. If $H$ is a subdivided star, a singleton or the line graph of a subdivided star, we say that $(H,X)$ is a concentrated connectifier.

The following was proved in [3]:

Theorem 5.1 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

For every integer $h\geq 1$ , there exists an integer $\nu=\nu(h)\geq 1$ with the following property. Let $G$ be a connected graph with no clique of cardinality $h$ . Let $S\subseteq G$ such that $|S|\geq\nu$ , $G\setminus S$ is connected and every vertex of $S$ has a neighbor in $G\setminus S$ . Then there is a set $\tilde{S}\subseteq S$ with $|\tilde{S}|=h$ and an induced subgraph $H$ of $G\setminus S$ for which one of the following holds.

•

$H$ is a path and every vertex of $\tilde{S}$ has a neighbor in $H$ ; or
•

$H$ is a caterpillar, or the line graph of a caterpillar, or a subdivided star. Moreover, every vertex of $\tilde{S}$ has a unique neighbor in $H$ and $H\cap N(\tilde{S})=\mathcal{Z}(H)$ .

We now prove a version of Theorem 5.1 that does not assume a bound on the clique number. This result may be of independent use in the future (See Figure 3 for a depiction of the outcomes).

Theorem 5.2.

For every integer $h\geq 1$ , there exists an integer $\mu=\mu(h)\geq 1$ with the following property. Let $G$ be a connected graph. Let $Y\subseteq G$ such that $|Y|\geq\mu$ , $G\setminus Y$ is connected and every vertex of $Y$ has a neighbor in $G\setminus Y$ . Then there is a set $Y^{\prime}\subseteq Y$ with $|Y^{\prime}|=h$ and an induced subgraph $H$ of $G\setminus Y$ for which one of the following holds.

•

$H$ is a path and every vertex of $Y^{\prime}$ has a neighbor in $H$ ; or
•

$H$ is a caterpillar, or the line graph of a caterpillar, or a subdivided star or the line graph of a subdivided star. Moreover, every vertex of $Y^{\prime}$ has a unique neighbor in $H$ and $H\cap N(Y^{\prime})=\mathcal{Z}(H)$ .

Proof.

Let $\mu(h)=\nu(h+1)$ where $\nu$ is as in Theorem 5.1. Let $F$ be a minimal connected subset of $G\setminus Y$ such that every $y\in Y$ has a neighbor in $F$ .

(5) If $F$ contains a clique of size $h$ , then there exists $H\subseteq F$ and $Y^{\prime}\subseteq Y$ with $|Y^{\prime}|=h$ such that $(H,Y^{\prime})$ is a clique connectifier.

Suppose that there is a clique $K$ of size $h$ in $F$ . It follows from the minimality of $F$ that for every $k\in K$ one of the following holds:

•

There is $y(k)\in Y$ such that $y(k)$ is anticomplete to $F\setminus k$ ; in this case set $C(k,y(k))=\emptyset$ .
•

$F\setminus\{k\}$ is not connected, and for every component $C$ of $F\setminus k$ there is $y\in Y$ such that $y$ is anticomplete to $F\setminus(C\cup\{k\})$ . Since $K$ is a clique, there is a component $C$ of $F\setminus k$ such that $K\cap C=\emptyset$ . Let $y(k)$ be a vertex of $Y$ that is anticomplete to $F\setminus(C\cup\{k\})$ ; write $C(k,y(k))=C$ .

Let $Y^{\prime}$ be the set of all vertices $y(k)$ as above. For every $k\in K$ , let $H_{k}$ be a path from $k$ to $y(k)$ with $H_{k}^{*}\subseteq C(k,y(k))$ . Write $H=\bigcup_{k\in K}H_{k}$ . We will show that $|Y^{\prime}|=h$ and $(H,Y^{\prime})$ is a clique connectifier. It follows from the definition of $H_{k}$ that every vertex of $Y^{\prime}$ has a neighbor in $H$ .

Next we claim that if $k\neq k^{\prime}$ , then $H_{k}$ is disjoint from and anticomplete to $H_{k^{\prime}}$ . Recall that $H_{k}\subseteq C(k,y(k))\cup\{k\}$ , and $y(k)$ is anticomplete to $F\setminus(C(k,y(k))\cup\{k\})$ . It follows that $H_{k}\cap K=\{k\}$ , and so $k\not\in H_{k^{\prime}}$ and $k^{\prime}\not\in H_{k}$ . Let $D$ be the component of $F\setminus k$ such that $k^{\prime}\in D$ . By the definition of $C(k,y(k))$ , we have that $D\neq C(k,y(k))$ . Since $H_{k^{\prime}}$ is connected and $k\not\in H_{k^{\prime}}$ , it follows that $H_{k^{\prime}}\subseteq D$ . Consequently, $H_{k}$ and $H_{k^{\prime}}$ are disjoint and anticomplete to each other. Since $y(k)$ has no neighbor in $D$ , we deduce that $y(k)$ is anticomplete to $H_{k^{\prime}}$ , and in particular $y(k)\neq y(k^{\prime})$ . Similarly, $y(k^{\prime})$ is anticomplete to $H_{k}$ . This proves the claim.

Now it follows from the claim that if $k\neq k^{\prime}$ , then $y(k)\neq y(k^{\prime})$ . Consequently, $|Y^{\prime}|=|K|=h$ , and $(H,Y^{\prime})$ is a clique connectifier. This proves (5).

By (5), we may assume that $F$ is $K_{h}$ -free. Since $S$ is a stable set, it follows that $F\cup S$ is $K_{h+1}$ -free. Now the result follows from Theorem 5.1 applied to $F\cup S$ . ∎

Now we move to two anticomplete connected subgraphs and prove the following:

Theorem 5.3.

For every integer $x\geq 1$ , there exists an integer $\phi=\phi(x)\geq 1$ with the following property. Let $G\in\mathcal{C}$ and assume that $V(G)=D_{1}\cup D_{2}\cup Y$ where

•

$Y$ is a stable set with $|Y|=\phi$ ,
•

$D_{1}$ and $D_{2}$ are components of $G\setminus Y$ , and
•

$N(D_{1})=N(D_{2})=Y$ .

Then there exist $X\subseteq Y$ with $|X|=x$ , $H_{1}\subseteq D_{1}$ and $H_{2}\subseteq D_{2}$ (possibly with the roles of $D_{1}$ and $D_{2}$ reversed) such that either:

(1)
Not both $(H_{1},X)$ and $(H_{2},X)$ are alignments, and
- •
  
  $(H_{1},X)$ is a triangular connectifier or a clique connectifier or a triangular alignment; and
- •
  
  $(H_{2},X)$ is a stellar connectifier, or a spiky connectifier, or a spiky alignment or a wide alignment.
or
(2)

Both $(H_{1},X)$ and $(H_{2},X)$ are alignments, and at least one of $(H_{1},X)$ and $(H_{2},X)$ is not a spiky alignment.

Moreover, if neither of $(H_{1},X),(H_{2},X)$ is a concentrated connectifier, then the orders given on $X$ by $(H_{1},X)$ and by $(H_{2},X)$ are the same.

In this paper we do not need the full generality of Theorem 5.3; we are only interested in two special cases: when $D_{1}$ is a path and when the clique number of $G$ is bounded. However, it is easier to prove the more general result first using the symmetry between $D_{1}$ and $D_{2}$ , and then use it to handle the special cases. We also believe that Theorem 5.3 will be useful in the future.

We start by recalling a well known theorem of Erdős and Szekeres [20].

Theorem 5.4 (Erdős and Szekeres [20]).

Let $x_{1},\dots,x_{n^{2}+1}$ be a sequence of distinct reals. Then there exists either an increasing or a decreasing $(n+1)$ -sub-sequence.

We start with two lemmas.

Lemma 5.5.

Let $G\in\mathcal{C}$ and assume that $V(G)=H_{1}\cup H_{2}\cup X$ where $X$ is a stable set with $|X|\geq 3$ and $H_{1}$ is anticomplete to $H_{2}$ . Suppose that for $i\in\{1,2\}$ , the pair $(H_{i},X)$ is a consistent alignment, or a consistent connectifier. Assume also that if neither of $(H_{1},X),(H_{2},X)$ is concentrated, then the orders given on $X$ by $(H_{1},X)$ and by $(H_{2},X)$ are the same. Then (possibly switching the roles of $H_{1}$ and $H_{2}$ ), we have that either:

(1)
Not both $(H_{1},X)$ and $(H_{2},X)$ are alignments, and
- •
  
  $(H_{1},X)$ is a triangular connectifier or a clique connectifier or a triangular alignment; and
- •
  
  $(H_{2},X)$ is a stellar connectifier, or a spiky connectifier, or a spiky alignment or a wide alignment.
or
(2)

Both $(H_{1},X)$ and $(H_{2},X)$ are alignments, and at least one of $(H_{1},X)$ and $(H_{2},X)$ is not a spiky alignment, and at least one of $(H_{1},X)$ and $(H_{2},X)$ is not a triangular alignment.

Proof.

If at least one $(H_{i},X)\subseteq\{(H_{1},X),(H_{2},X)\}$ is not a concentrated connectifier, we let $x_{1},\dots,x_{k}$ be the order given on $X$ by $(H_{i},X)$ . If both of $(H_{i},X)$ are concentrated connectifiers, we let $x_{1},\dots,x_{k}$ be an arbitrary order on $X$ . Let $H$ be the unique hole contained in $H_{1}\cup H_{2}\cup\{x_{1},x_{k}\}$ . For $j\in\{1,2\}$ and $i\in\{1,\dots,k\}$ , if $H_{j}$ is a connectifier, let $D_{i}^{j}$ be the leg of $H_{j}$ containing a neighbor of $x_{i}$ ; and if $H_{j}$ is an alignment let $D_{i}^{j}=\emptyset$ .

Suppose first that $(H_{1},X)$ is a triangular alignment, a clique connectifier or a triangular connectifier. If $(H_{2},X)$ is a triangular alignment, a clique connectifier or a triangular connectifier, then for every $i\in\{2,\dots,k-1\}$ , the graph $H\cup D_{i}^{1}\cup D_{i}^{2}\cup\{x_{i}\}$ is either a prism or an even wheel with center $x_{i}$ , a contradiction. This proves that $(H_{2},X)$ is a stellar connectifier, or a spiky connectifier, or a spiky alignment, or a wide alignment, as required.

Thus we may assume that for $i\in\{1,2\}$ , the pair $(H_{i},X)$ is a stellar connectifier, or a spiky connectifier, or a spiky alignment, or a wide alignment. If $(H_{1},X)$ is a stellar connectifier or a spiky connectifier, then for every $x_{i}\in X\setminus\{x_{1},x_{k}\}$ , the graph $H\cup D_{i}^{1}\cup D_{i}^{2}\cup\{x_{i}\}$ contains a theta, a contradiction.

It follows that for $i\in\{1,2\}$ , the pair $(H_{i},X)$ is an alignment. We may assume that $(H_{1},X)$ and $(H_{2},X)$ are either both spiky alignments or both triangular alignments, for otherwise the second outcome of the theorem holds. But now for every $x_{i}\in X\setminus\{x_{1},x_{k}\}$ , the graph $H\cup\{x_{i}\}$ is either a theta or an even wheel, a contradiction. ∎

Lemma 5.6.

Let $G$ be a theta-free graph and let $P=p_{1}\hbox{-}\cdots\hbox{-}p_{k}$ be a path in $G$ . Let $D$ be a connected subset of $G$ such that $D$ is anticomplete to $P$ . Let $x,y\in N(P)\cap N(D)$ such that $xy$ is not an edge. Assume that each of $x$ and $y$ has at least two non-adjacent neighbors in $P$ . Let $i_{x}$ be minimum and $j_{x}$ be maximum such that $x$ is adjacent to $p_{i_{x}}$ and $p_{j_{x}}$ and write $P_{x}=p_{i_{x}}\hbox{-}P\hbox{-}p_{j_{x}}$ . Suppose that $y$ has a neighbor in $P_{x}$ . Then $y$ has a neighbor in $N_{P_{x}}[p_{i_{x}}]\cup N_{P_{x}}[p_{j_{x}}]$ .

Proof.

Suppose that $y$ is anticomplete to $N_{P}[p_{i_{x}}]\cup N_{P}[p_{j_{x}}]$ . Let $R$ be a path from $x$ to $y$ with $R^{*}\subseteq D$ . (See Figure 4.) Let $i\in\{{i_{x}},\dots,{j_{x}}\}$ be minimum and $j\in\{{i_{x}},\dots,{j_{x}}\}$ be maximum such that $y$ is adjacent to $p_{i},p_{j}$ . Then there there is a path $P_{i}$ from $x$ to $y$ with interior in $p_{i_{x}}\hbox{-}P\hbox{-}p_{i}$ , and a path $P_{j}$ from $x$ to $y$ with interior in $p_{j}\hbox{-}P\hbox{-}p_{j_{x}}$ . If $j>i+1$ , then $P_{i}\cup P_{j}\cup R$ is a theta with ends $x,y$ , a contradiction; therefore $j\leq i+1$ . Since $y$ has at least two non-adjacent neighbors in $P$ , it follows that $y$ has a neighbor $p_{z}$ in $P\setminus P_{x}$ . We may assume that $z>j_{x}$ , and therefore there is a path $Q$ from $x$ to $y$ with $Q^{*}\subseteq p_{j_{x}}\hbox{-}P\hbox{-}p_{z}$ . Since $y$ is anticomplete to $N_{P_{x}}[p_{i_{x}}]\cup N_{P_{x}}[p_{j_{x}}]$ , we have that $j<j_{x}-1$ , and so $P_{i}^{*}$ is anticomplete to $Q^{*}$ . But now $P_{i}\cup Q\cup R$ is a theta with ends $x,y$ , a contradiction. ∎

We can now prove Theorem 5.3. The proof follows along the lines of [3], but the assumptions are different.

Proof.

Let $z=12^{2}36^{2}x^{6}$ and let $\phi(x)=\mu(\mu(z))$ , where $\mu$ is as in Theorem 5.2. Applying Theorem 5.2 twice, we obtain a set $Z\subseteq Y$ with $|Z|=z$ and $H_{i}\subseteq D_{i}$ such that either

•

$H_{i}$ is a path and every vertex of $Z$ has a neighbor in $H_{i}$ ; or
•

$(H_{i},X)$ is a consistent connectifier

for every $i\in\{1,2\}$ .

(6) Let $i\in\{1,2\}$ and $y\in\mathbb{N}$ . If $H_{i}$ is a path and every vertex of $Z$ has a neighbor in $H_{i}$ , then either some vertex of $H_{i}$ has $y$ neighbors in $Z$ , or there exists $Z^{\prime}\subseteq Z$ with $|Z^{\prime}|\geq\frac{|Z|}{12y}$ and a subpath $H_{i}^{\prime}$ of $H_{i}$ such that $(H_{i}^{\prime},Z^{\prime})$ is a consistent alignment.

Let $H_{i}=h_{1}\hbox{-}\cdots\hbox{-}h_{k}$ . We may assume that $H_{i}$ is chosen minimal satisfying Theorem 5.2, and so there exist $z_{1},z_{k}\in Z$ such that $N_{H_{i}}(z_{j})=\{h_{j}\}$ for $j\in\{1,k\}$ .

We may assume that $|N_{Z}(h)|<y$ for every $h\in H_{i}$ . Let $Z_{1}$ be the set of vertices in $Z$ with exactly one neighbor in $H_{i}$ . Suppose that $|Z_{1}|\geq\frac{|Z|}{3}$ . It follows that $Z_{1}$ contains a set $Z^{\prime}$ with $|Z^{\prime}|\geq\frac{|Z_{1}|}{y}\geq\frac{|Z|}{3y}$ such that no two vertices in $Z^{\prime}$ have a common neighbor in $H_{i}$ . We may assume that $z_{1},z_{k}\in Z^{\prime}$ . Therefore, $(H_{i},Z^{\prime})$ is a spiky alignment, as required. Thus we may assume that $|Z_{1}|<\frac{|Z|}{3}$ .

Next, let $Z_{2}$ be the set of vertices in $z\in Z$ such that either $z\in\{z_{1},z_{k}\}$ or $z$ has exactly two neighbors in $H_{i}$ , and they are adjacent. Suppose that $|Z_{2}|\geq\frac{|Z|}{3}$ . By choosing $Z^{\prime}$ greedily, it follows that $Z_{2}$ contains a subset $Z^{\prime}$ with the following specifications:

•

$z_{1},z_{k}\in Z^{\prime}$ ;
•

$|Z^{\prime}|\geq\frac{|Z_{2}|}{2y}\geq\frac{|Z|}{6y}$ ; and
•

no two vertices in $Z^{\prime}$ have a common neighbor in $H_{i}$ .

But then $(H_{i},Z^{\prime})$ is a triangular alignment, as required. Thus we may assume that $|Z_{2}|<\frac{|Z|}{3}$ .

Finally, let $Z_{3}=\{z_{1},z_{k}\}\cup(Z\setminus(Z_{1}\cup Z_{2}))$ . It follows that $|Z_{3}|\geq\frac{|Z|}{3}$ . Let $R$ be a path from $z_{1}$ to $z_{k}$ with $R^{*}\subseteq H_{3-i}$ , and let $H$ be the hole $z_{1}\hbox{-}H_{i}\hbox{-}z_{k}\hbox{-}R\hbox{-}z_{1}$ . Let $z\in Z_{3}\setminus\{z_{1},z_{k}\}$ . Define $H_{i}(z)$ to be the minimal subpath of $H_{i}$ containing $N_{H_{i}}(z)$ . Let the ends of $H_{i}(z)$ be $a_{z}$ and $b_{z}$ . Next let $Bad(z)=N_{H_{i}(z)}[a_{z},b_{z}]$ . Since $H_{i}$ is a path, it follows that $|Bad(z)|\leq 4$ for every $z$ . Since $N_{Z}(h)<y$ for every $h\in H_{i}$ , we can greedily choose $Z^{\prime}\subseteq Z_{3}$ with $|Z^{\prime}|\geq\frac{|Z_{3}|}{4y}\geq\frac{|Z|}{12y}$ , where $z_{1},z_{k}\in Z^{\prime}$ and such that if $z,z^{\prime}\in Z^{\prime}$ , then $z^{\prime}$ is anticomplete to $Bad(z)$ . It follows from Lemma 5.6 that $H_{i}(z)\cap H_{i}(z^{\prime})=\emptyset$ for every $z,z^{\prime}\in Z^{\prime}$ , and so $(H_{i},Z^{\prime})$ is a wide alignment. This proves (5).

(7) There exist $Z^{\prime}\subseteq Z$ with $|Z^{\prime}|\geq x^{2}$ , and $H_{i}^{\prime}\subseteq H_{i}$ for $i=1,2$ such that $(H_{i}^{\prime},Z^{\prime})$ is a consistent alignment or a consistent connectifier.

If both $(H_{1},Z)$ and $(H_{2},Z)$ are consistent connectifiers, (5) holds. Thus we may assume that $H_{1}$ is a path and every vertex of $Z$ has a neighbor in $H_{1}$ . Suppose first that some $h\in H_{1}$ has at least $36x^{2}$ neighbors in $Z$ . Let $H_{1}^{\prime}=\{h\}$ , and let $Z^{\prime\prime}\subseteq Z\cap N(h)$ with $|Z^{\prime\prime}|=36x^{2}$ . If $(H_{2},Z)$ is a connectifier, we can clearly choose $H_{2}^{\prime}\subseteq H_{2}$ such that (5) holds. So we may assume that $H_{2}$ is a path and every vertex of $Z$ has a neighbor in $H_{2}$ . Moreover, no vertex $f$ of $H_{2}$ has three or more neighbors in $Z^{\prime}$ , for otherwise $\{f,h\}\cup(N(f)\cap Z^{\prime})$ contains a theta with ends $h,f$ . Now by (5) applied with $y=3$ , there is a set $Z^{\prime\prime}\subseteq Z^{\prime}$ with $|Z^{\prime\prime}|\geq x^{2}$ such that $(H_{2},Z^{\prime\prime})$ is a consistent alignment, and (5) holds. Similarly, we may assume that every $h\in H_{2}$ has fewer than $36x^{2}$ neighbors in $Z$ .

Therefore, we may assume that every vertex $h\in H_{1}$ has strictly fewer than $36x^{2}$ neighbors in $Z$ . Applying (5) with $y=36x^{2}$ , we conclude that there exists $Z_{1}\subseteq Z$ with $|Z_{1}|\geq 36\times 12x^{4}$ and a path $H_{1}^{\prime}\subseteq H_{1}$ such that $(H_{1}^{\prime},Z_{1})$ is a consistent alignment. If $(H_{2},Z)$ is a consistent connectifier, then (5) holds, so we may assume that $H_{2}$ is a path and every vertex of $Z$ has a neighbor in $H_{2}$ . Since every vertex $h\in H_{2}$ has fewer than $36x^{2}$ neighbors in $Z$ , we apply (5) with $y=36x^{2}$ again, and we conclude that there exists $Z^{\prime}\subseteq Z_{1}$ with $|Z^{\prime}|\geq x^{2}$ and a path $H_{2}^{\prime}\subseteq H_{2}$ such that $(H_{2}^{\prime},Z^{\prime})$ is a consistent alignment. This proves (5).

(8) There exist $\hat{Z}\subseteq Z$ with $|\hat{Z}|\geq x$ , and $\hat{H_{i}}\subseteq H_{i}$ for $i=1,2$ such that $(\hat{H}_{i},\hat{Z})$ is a consistent alignment or a consistent connectifier. Moreover, if neither of $(\hat{H_{1}},\hat{Z)},(\hat{H_{2}},\hat{Z})$ is a concentrated connectifier, then the order given on $\hat{Z}$ by $(\hat{H_{1}},\hat{Z})$ and $(\hat{H_{2}},\hat{Z})$ is the same.

Let $Z^{\prime},H_{1}^{\prime},H_{2}^{\prime}$ be as in (5). We may assume that neither of $(H_{1}^{\prime},Z^{\prime})$ and $(H_{2}^{\prime},Z^{\prime})$ is a concentrated connectifier. For $i\in\{1,2\}$ , let $\pi_{i}$ be the order given on $Z^{\prime}$ by $(H_{i}^{\prime},Z^{\prime})$ . By Theorem 5.4 there exists $\hat{Z}\subseteq Z^{\prime}$ such that (possibly reversing $H_{i}^{\prime}$ ) the orders $\pi_{i}$ restricted to $\hat{Z}$ are the same, as required. This proves (5).

Now Theorem 5.3 follows from Lemma 5.5. ∎

We now refine Theorem 5.3 in the two special cases we need here. The first one is when $D_{1}$ is a path. It is useful to state this result in terms of the following definition.

Given a graph class $\mathcal{G}$ , we say $\mathcal{G}$ is amiable if, for every integer $x\geq 2$ , there exists an integer $\sigma=\sigma(x)\geq 1$ such that the following holds for every graph $G\in\mathcal{G}$ . Assume that $V(G)=D_{1}\cup D_{2}\cup Y$ where

•

$Y$ is a stable set with $|Y|=\sigma$ ,
•

$D_{1}$ and $D_{2}$ are components of $G\setminus Y$ ,
•

$N(D_{1})=N(D_{2})=Y$ ,
•

$D_{1}=d_{1}\hbox{-}\cdots\hbox{-}d_{k}$ is a path, and
•

for every $y\in Y$ there exists $i(y)\in\{1,\dots,k\}$ such that $N(d_{i(y)})\cap Y=\{y\}$ .

Then there exist $X\subseteq Y$ with $|X|=x+2$ , $H_{1}\subseteq D_{1}$ and $H_{2}\subseteq D_{2}$ such that

(1)
One of the following holds:
- •
  
  $(H_{1},X)$ is a triangular alignment and $(H_{2},X)$ is a stellar connectifier or a spiky connectifier;
- •
  
  $(H_{1},X)$ is a spiky alignment, and $(H_{2},X)$ is a triangular connectifier or a clique connectifier;
- •
  
  $(H_{1},X)$ is a wide alignment and $(H_{2},X)$ is a triangular connectifier or a clique connectifier; or
- •
  
  Both $(H_{1},X)$ and $(H_{2},X)$ are alignments, and at least one of $(H_{1},X)$ and $(H_{2},X)$ is not a spiky alignment, and at least one of $(H_{1},X)$ and $(H_{2},X)$ is not a triangular alignment.
(2)

If $(H_{2},X)$ is not a concentrated connectifier, then the orders given on $X$ by $(H_{1},X)$ and by $(H_{2},X)$ are the same.
(3)

For every $x\in X$ except at most two, we have that $N_{D_{1}}(x)=N_{H_{1}}(x)$ .

Theorem 5.7.

The class $\mathcal{C}$ is amiable.

Proof.

Let $x\geq 2$ , and let $G\in\mathcal{C},D_{1}=d_{1}\hbox{-}\cdots\hbox{-}d_{k},D_{2},Y$ be as in the definition of an amiable class. Let $\sigma(x)=\phi(x+4)$ where $\phi$ is as in Theorem 5.3. We start with the following.

(9) For every $d\in D_{1}$ , we have that $|N_{Y}(d)|\leq 4$ .

Suppose there exists $i\in\{1,\dots,k\}$ and $y_{1},y_{2},y_{3},y_{4},y_{5}\in Y\cap N(d_{i})$ . We may assume that $i(y_{1})<\dots<i(y_{5})$ . By reversing $D_{1}$ if necessary, we may assume that $i(y_{3})<i$ , and so $i(y_{2})<i-1$ . Let $P$ be a path with ends $y_{1},y_{2}$ and with interior in $d_{i(y_{1})}\hbox{-}D_{1}\hbox{-}d_{i(y_{2})}$ . Let $R$ be a path from $y_{1}$ to $y_{2}$ with interior in $D_{2}$ . Now there is a theta in $G$ with ends $y_{1},y_{2}$ and paths $P$ , $R$ and $y_{1}\hbox{-}d_{i}\hbox{-}y_{2}$ , a contradiction. This proves (5).

Now we apply Theorem 5.3 and obtain a set $X^{\prime}$ with $|X^{\prime}|=x+4$ satisfying one of the outcomes of Theorem 5.3. Let $(H_{1},X^{\prime})$ and $(H_{2},X^{\prime})$ be as in Theorem 5.3 where $H_{1}\subseteq D_{1}$ (and so Theorem 5.3 may hold with the roles of $H_{1}$ and $H_{2}$ reversed). It follows from (5) that every vertex in $D_{1}$ has degree at most 6 in $G$ and degree 2 in $G[D_{1}]$ , and so $(H_{1},X^{\prime})$ is an alignment. Let $z_{1},\dots,z_{x+4}$ be the order on $X^{\prime}$ given by $(H_{1},X^{\prime})$ .

(10) There exist at most two values of $i\in\{2,\dots,x+3\}$ for which $N(z_{i})\cap D_{1}\neq N(z_{i})\cap H_{1}$ .

Suppose there exist three such values of $i$ . Since $H_{1}$ is a path, there exist $i,j\in\{1,\dots,k\}$ such that $H_{1}=d_{i}\hbox{-}D_{1}\hbox{-}d_{j}$ , and we may assume (by reversing $D_{1}$ if necessary) that for two values $p,q\in\{2,\dots,x+3\}$ , both $z_{p}$ and $z_{q}$ have neighbors in $d_{1}\hbox{-}D_{1}\hbox{-}d_{i-1}$ . Let $P$ be a path from $z_{p}$ to $z_{q}$ with $P^{*}\subseteq d_{1}\hbox{-}D_{1}\hbox{-}d_{i-1}$ . Since $p,q>1$ , there is a path $Q$ from $z_{p}$ to $z_{q}$ with $Q^{*}\subseteq H_{1}\setminus d_{i}$ . But now we get a theta with ends $z_{p},z_{q}$ and path $P$ , $Q$ , and a path from $z_{p}$ to $z_{q}$ with interior in $D_{2}$ , a contradiction. This proves (5).

By (5), there exists $X\subseteq\{z_{2},\dots,z_{x+3}\}$ with $|X|=x$ and such that $N_{D_{1}}(z)=N_{H_{1}}(z)$ for every $z\in Z$ . We obtain that $(H_{1},X\cup\{z_{1},z_{x+4}\})$ and $(H_{2},X\cup\{z_{1},z_{x+4}\})$ satisfy the first statement of Theorem 5.7. The second statement of Theorem 5.7 follows immediately from Theorem 5.3, and the third statement holds by the choice of $X$ . ∎

The second special case is when the clique number of $G$ is bounded, and $Y\cap\operatorname{Hub}(G)=\emptyset$ ; it is a result from [3].

Theorem 5.8 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

For every pair of integers $t,x\geq 1$ , there exists an integer $\tau=\tau(t,x)\geq 1$ with the following property. Let $G\in\mathcal{C}_{t}$ and assume that $V(G)=D_{1}\cup D_{2}\cup Y$ where

•

$Y$ is a stable set with $|Y|=\tau$ ,
•

$D_{1}$ and $D_{2}$ are components of $G\setminus Y$ , and
•

$N(D_{1})=N(D_{2})=Y$ .

Assume that $Y\cap\operatorname{Hub}(G)=\emptyset$ . Then there exist $X\subseteq Y$ with $|X|=x$ , $H_{1}\subseteq D_{1}$ and $H_{2}\subseteq D_{2}$ (possibly with the roles of $D_{1}$ and $D_{2}$ reversed) such that

•

$(H_{1},X)$ is a triangular connectifier or a triangular alignment;
•

$(H_{2},X)$ is a stellar connectifier, or a spiky connectifier, or a spiky alignment or a wide alignment; and
•

if $(H_{1},X)$ is a triangular alignment, then $(H_{2},X)$ is not a wide alignment.

Moreover, if neither of $(H_{1},X),(H_{2},X)$ is a stellar connectifier, then the orders given on $X$ by $(H_{1},X)$ and by $(H_{2},X)$ are the same.

6. Bounding the number of non-hubs in a minimal separator

The goal of this section is to prove Theorem 6.3. This is the second main ingredient of the proof of Theorem 1.3 (in addition to the machinery of Section 4). Theorem 6.3 is what allows us to iterate the construction of Section 4 for $\mathcal{O}(\log|V(G)|)$ rounds (rather than just constantly many), which is what we need in the proof. We need the following definition and result. Given a graph $H$ , a $(\leq p)$ -subdivision of $H$ is a graph that arises from $H$ by replacing each edge $uv$ of $H$ by a path from $u$ to $v$ of length at least 1 and at most $p$ ; the interiors of these paths are pairwise disjoint and anticomplete.

Theorem 6.1 (Lozin and Razgon [29]).

For all positive integers $p$ and $r$ , there exists a positive integer $m=m(p,r)$ such that every graph $G$ containing a ( $\leq p$ )-subdivision of $K_{m}$ as a subgraph contains either $K_{p,p}$ as a subgraph or a proper ( $\leq p$ )-subdivision of $K_{r,r}$ as an induced subgraph.

Since graphs in $\mathcal{C}_{t}$ contain neither $K_{t+1}$ nor $K_{2,2}$ as an induced subgraph, we conclude that graphs in $\mathcal{C}_{t}$ do not contain $K_{t+1,t+1}$ as a subgraph. Likewise, graphs in $\mathcal{C}_{t}$ do not contain subdivisions of $K_{2,3}$ (in other words, thetas) as an induced subgraph. Therefore, letting $m=m(t+1,3)$ , we conclude:

Lemma 6.2.

Let $t\in\mathbb{N}$ . Then there exists an $m=m(t)\in\mathbb{N}$ such that the following holds: Let $G$ in $\mathcal{C}_{t}$ . Then $G$ does not contain a $(\leq 2)$ -subdivision of $K_{m}$ as a subgraph.

Recall that the Ramsey number $R(t,s)$ is the minimum integer such that every graph on at least $R(t,s)$ vertices contains either a clique of size $t$ or a stable set of size $s$ .

Theorem 6.3.

For every integer $t\geq 1$ there exists $\Gamma=\Gamma(t)$ with the following property. Let $G\in\mathcal{C}_{t}$ and assume that $V(G)=D_{1}\cup D_{2}\cup Y$ where

•

$D_{1}$ and $D_{2}$ are components of $G\setminus Y$ .
•

$Y$ is a stable set.
•

$N(D_{1})=N(D_{2})=Y$ .
•

$Y\cap\operatorname{Hub}(G)=\emptyset$ .
•

There exist $a_{1}\in D_{1}$ and $a_{2}\in D_{2}$ such that $a_{1}a_{2}$ is a $|Y|$ -banana in $G$ .

Then $|Y|\leq\Gamma$ .

Proof.

Let $k=k(t)$ be as in Theorem 3.2. We may assume that $k\geq 3$ . Let $m=m(t)$ as in Lemma 6.2. Let $\lambda=R(m,k).$ Let $\tau=\tau(t,\lambda)$ be as in Theorem 5.8. Set $\Gamma=\tau$ . Applying Theorem 5.8, we deduce that there exist $H_{1}\subseteq D_{1}$ , $H_{2}\subseteq D_{2}$ and $X\subseteq Y$ with $|X|=\lambda$ such that:

•

$(H_{1},X)$ is a triangular connectifier or a triangular alignment;
•

$(H_{2},X)$ is a stellar connectifier, or a spiky connectifier, or a spiky alignment, or a wide alignment; and
•

if $(H_{1},X)$ is a triangular alignment, then $(H_{2},X)$ is not a wide alignment.

(11) Suppose $(H_{2},X)$ is a wide alignment, and let $x_{1},\dots,x_{\lambda}$ be the order on $X$ given by $H_{2}$ . Then for every $i\in\{2,\dots,\lambda-1\}$ , $x_{i}$ has exactly three neighbors in $H_{2}$ , and two of them are adjacent.

Let $i\in\{2,\dots,\lambda-1\}$ . Let $R$ be a path from $x_{1}$ to $x_{\lambda}$ with interior in $H_{1}$ . Then $H=x_{1}\hbox{-}H_{2}\hbox{-}x_{\lambda}\hbox{-}R\hbox{-}x_{1}$ is a hole. Since $(H_{2},X)$ is a wide alignment, $x_{i}$ has two non-adjacent neighbors in $H$ . Since $(H,x_{i})$ is not a wheel in $G$ , and $x_{i}$ is non-adjacent to $x_{1}$ and $x_{\lambda}$ , (6) follows.

Next we show:

(12) Every vertex in $D_{1}$ has at most two neighbors in $X$ .

Suppose that there is a vertex $d\in D_{1}$ such that $d$ has three distinct neighbors $x_{i},x_{j},x_{k}$ in $X$ ; without loss of generality, we may assume that $i<j<k$ . We consider two cases. First, if $H_{2}$ is a wide alignment, then $G$ contains a wheel $(C,x_{j})$ where $C$ is a cycle consisting of $x_{i},d,x_{k}$ and the path from $x_{k}$ to $x_{i}$ with interior in $H_{2}$ . By (6), $x_{j}$ has four neighbors in $C$ : $d$ , as well as three neighbors in $H_{2}$ . But $x_{j}$ is not a hub in $G$ , a contradiction.

It follows that $H_{2}$ is not a wide alignment. Now, for $r,s\in\{i,j,k\}$ , let $P_{rs}$ be a path from $x_{r}$ to $x_{s}$ with interior in $H_{2}$ . Then $\{d\}\cup P_{ij}\cup P_{jk}\cup P_{ik}$ is a theta in $G$ (see Figure 5). This proves (6).

(13) There exists $S\subseteq X$ with $|S|=k$ , such that for every $s,s^{\prime}\in S$ and for every path $P$ from $s$ to $s^{\prime}$ with interior in $D_{1}$ we have that $|E(P)|>2$ .

Define a graph $H$ with vertex set $X$ such that $xx^{\prime}\in E(H)$ if and only if there is a path of length two with ends $x,x^{\prime}$ and interior in $D_{1}$ . Since $|X|=\lambda=R(m,k)$ , there is $S\subseteq X$ with $|S|=k$ such that $S$ is either a clique or a stable set in $H$ .

If $S$ is a stable set, then (6) holds, so we may assume that $S$ is a clique of size $m$ in $H$ . Write $S=\{s_{1},\dots,s_{m}\}$ . For $i,j\in\{1,\dots,m\}$ with $i\neq j$ , we let $d_{ij}$ be a common neighbor of $s_{i}$ and $s_{j}$ in $D_{1}$ (which exists, since $S$ is a clique in $H$ ). By (6), no vertex has three neighbors in $S$ , and so the vertices $d_{ij}$ are pairwise distinct. But now $S\cup\{d_{ij}:i,j\in\{1,\dots,m\},i\neq j\}$ contains a $(\leq 2)$ -subdivision of $K_{m}$ as a subgraph, contrary to Lemma 6.2. This is proves (6).

From now on let $S\subseteq X$ be as in (6). Let $G^{\prime}$ be the graph obtained from $D_{1}\cup S$ by adding a new vertex $v$ with $N(v)=S$ . We need the following two facts about $G^{\prime}$ :

(14) $S\cap\operatorname{Hub}(G^{\prime})=\emptyset$ .

Suppose not, let $s\in S$ and let $(H,s)$ be a wheel in $G^{\prime}$ . Since $S\cap\operatorname{Hub}(G)=\emptyset$ , it follows that $v\in H$ . Let $N_{H}(v)=\{s^{\prime},s^{\prime\prime}\}$ ; then $H\setminus v$ is a path $R$ from $s^{\prime}$ to $s^{\prime\prime}$ with $R^{*}\subseteq D_{1}$ . Since $(H,s)$ is a wheel, it follows that $s$ has two non-adjacent neighbors in $R$ . Consequently, there is a path $P^{\prime}$ from $s$ to $s^{\prime}$ and a path $P^{\prime\prime*}$ form $s$ to $s^{\prime\prime}$ , both with interior in $R$ , such that $P^{\prime*}$ is disjoint from and anticomplete to $P^{\prime\prime*}$ .

Let $T$ be a path from $s^{\prime}$ to $s^{\prime\prime}$ with interior in $H_{2}$ . Then $H^{\prime}=s^{\prime}\hbox{-}R\hbox{-}s^{\prime\prime}\hbox{-}T\hbox{-}s^{\prime}$ is a hole. Since $X\cap\operatorname{Hub}(G)=\emptyset$ , it follows that $(H^{\prime},s)$ is not a wheel in $G$ , and so $s$ is anticomplete to $T$ .

Suppose first that $(H_{2},X)$ is a wide alignment. Since $s$ is anticomplete to $T$ , it follows that (reversing the order on $X$ if necessary, and exchanging the roles of $s^{\prime},s^{\prime\prime}$ if necessary) $s,s^{\prime},s^{\prime\prime}$ appear in this order in the order given by $H_{2}$ on $X$ . Now we get a theta with ends $s,s^{\prime}$ and paths $s\hbox{-}P^{\prime}\hbox{-}s^{\prime}$ , $s\hbox{-}P^{\prime\prime}\hbox{-}s^{\prime\prime}\hbox{-}H_{2}\hbox{-}s^{\prime}$ , and $s\hbox{-}H_{2}\hbox{-}s^{\prime}$ , a contradiction. This proves that $(H_{2},X)$ is not a wide alignment.

Since $(H_{2},X)$ is not a wide alignment, there is a vertex $a\in H_{2}$ , and three paths $Q,Q^{\prime},Q^{\prime\prime}$ all with interior in $H_{2}$ , where $Q$ is from $a$ to $s$ , $Q^{\prime}$ is from $a$ to $s^{\prime}$ , and $Q^{\prime\prime}$ is from $a$ to $s^{\prime\prime}$ , and the sets $Q\setminus a$ , $Q^{\prime}\setminus a$ and $Q^{\prime\prime}\setminus a$ are pairwise disjoint and anticomplete to each other. Note that $T=s^{\prime}\hbox{-}Q^{\prime}\hbox{-}a\hbox{-}Q^{\prime\prime}\hbox{-}s^{\prime\prime}$ . Since $s$ is anticomplete to $T$ , it follows that $s$ is non-adjacent to $a$ . But now we get a theta with ends $a,s$ and path $s\hbox{-}Q\hbox{-}a$ , $s\hbox{-}P^{\prime}\hbox{-}s^{\prime}\hbox{-}Q^{\prime}\hbox{-}a$ , and $s\hbox{-}P^{\prime\prime}\hbox{-}s^{\prime\prime}\hbox{-}Q^{\prime\prime}\hbox{-}a$ , a contradiction. This proves (6).

(15) $G^{\prime}\subseteq\mathcal{C}_{t}$ .

Suppose not. Let $Z\subseteq G^{\prime}$ be such that $Z$ is a $K_{t}$ , a $C_{4}$ , a theta, a prism or an even wheel. Then $v\in Z$ . Since $N_{Z}(v)$ is a stable set, it follows that $Z$ is not a $K_{t}$ . Suppose first that $|N_{Z}(v)|=2$ ; then $N_{Z}(v)=\{s,s^{\prime}\}\subseteq S$ . Suppose that $Z$ is a $C_{4}$ . Then there is $d\in D_{1}$ such that $Z=v\hbox{-}s\hbox{-}d\hbox{-}s^{\prime}\hbox{-}v$ , contrary to the fact that $S$ was chosen as in (6). It follows that $Z$ is a theta, a prism or an even wheel. Let $Q$ be a path from $s$ to $s^{\prime}$ with interior in $D_{2}$ . Now $Z^{\prime}=(Z\setminus v)\cup Q$ is an induced subgraph of $G$ . Moreover, if $Z$ is a theta then $Z^{\prime}$ is a theta, and if $Z$ is a prism then $Z^{\prime}$ is a prism. Since $G\in\mathcal{C}$ , it follows that $Z$ is an even wheel; denote it by $(H,w)$ . Then $v\in H$ . By (6), $w\not\in S$ . It follows that $N_{Z}(w)=N_{Z^{\prime}}(w^{\prime})$ , and so $Z^{\prime}$ is an even wheel, contrary to the fact that $G\in\mathcal{C}$ .

This proves that $|N_{Z}(v)|>2$ . Since $N_{Z}(v)$ is a stable set, it follows that $Z$ is not a prism. Suppose that $Z$ is a theta. Then the ends of $Z$ are $v$ and $d\in D_{1}$ ; let the paths of $Z$ be $Z_{1},Z_{2},Z_{3}$ where $N_{Z_{i}}(v)=s_{i}$ . By renumbering $Z_{1},Z_{2},Z_{3}$ , we may assume that $s_{1},s_{2},s_{3}$ appear in this order in the order on $X$ given by $H_{1}$ . Then for every $i,j\in\{1,\dots,3\}$ , there is a unique path $F^{1}_{ij}$ from $s_{i}$ to $s_{j}$ with interior in $H_{1}$ . Similarly, for all $i,j\in\{1,\dots,3\}$ there is a unique path $F^{2}_{ij}$ from $s_{i}$ to $s_{j}$ with interior in $H_{2}$ . Observe that $F^{1}_{12}\cup F^{1}_{13}\cup F^{1}_{23}\cup F^{2}_{12}\cup F^{2}_{13}\cup F^{2}_{23}$ contains a unique pyramid $\Sigma$ with the following specifications:

•

The paths of $\Sigma$ are $P_{1},P_{2},P_{3}$ , where $s_{i}\in P_{i}$ .
•

Denote the apex of $\Sigma$ by $a$ , and let the base of $\Sigma$ be $b_{1}b_{2}b_{3}$ where $b_{i}$ is an end of $P_{i}$ . Then either $a\in H_{2}$ , or $(H_{2},X)$ is a wide alignment and $a=s_{2}$ .
•

$b_{1},b_{3}\in H_{1}$ , and $b_{2}\in H_{1}\cup\{s_{2}\}$ .

Let $\Sigma_{2}=\Sigma\setminus D_{1}$ . If $d$ is non-adjacent to $a$ , then $(Z\setminus v)\cup\Sigma_{2}$ is a theta with ends $a,d$ and paths $d\hbox{-}Z_{i}\hbox{-}s_{i}\hbox{-}P_{i}\hbox{-}a$ , contrary to the fact that $G\in\mathcal{C}_{t}$ . This proves that $d$ is adjacent to $a$ . Consequently, $a=s_{2}$ and $(H_{2},X)$ is a wide alignment. But now $H=s_{1}\hbox{-}Z_{1}\hbox{-}d\hbox{-}Z_{3}\hbox{-}s_{3}\hbox{-}F^{2}_{13}\hbox{-}s_{1}$ is a hole, and, since $s_{2}$ is adjacent to $d$ , $(H,s_{2})$ is a wheel, contrary to the fact that $S\cap\operatorname{Hub}(G)=\emptyset$ . This proves that $Z$ is not a theta.

Consequently, $Z$ is an even wheel $(H,w)$ . Since $|N_{Z}(v)|>2$ , it follows that either $v=w$ , or $v$ is adjacent to $w$ . If $v$ is adjacent to $w$ , then $w\in S$ , contrary to (6). It now follows that $w=v$ , and so $H$ is a hole in $D_{1}\cup S$ , where $|S\cap H|=l\geq 3$ . Let $S\cap H=\{s_{1},\dots,s_{l}\}$ where $s_{1},\dots s_{l}$ appear in this order in the order given on $S$ by $(H_{2},X)$ . If $(H_{2},X)$ is a spiky alignment or a wide alignment, or a spiky connectifier, then we get a theta with ends $s_{1},s_{2}$ two of whose paths are contained in $H$ , and the third is the path from $s_{1}$ to $s_{2}$ with interior in $H_{2}$ . It follows that $(H_{2},X)$ is a stellar connectifier. Therefore there exist paths $Q_{1},\dots,Q_{l}$ , all with a common end $a\in H_{2}$ , such that $Q_{i}$ is from $a$ to $s_{i}$ , $Q_{i}^{*}\subseteq H_{2}$ , and the sets $Q_{1}\setminus a,\dots,Q_{k}\setminus a$ are pairwise disjoint and anticomplete to each other. Since $(H,a)$ is a not an even wheel in $G$ , we deduce that $a$ is not complete to $\{s_{1},\dots,s_{l}\}$ ; let $s_{i}\in\{s_{1},\dots,s_{l}\}$ be non-adjacent to $a$ . Since $l\geq 3$ , there exist distinct $p,q\in\{1,\dots,l\}\setminus\{i\}$ and paths $R_{q}$ and $R_{p}$ of $H$ such that

•

$R_{p}$ is from $s_{i}$ to $s_{p}$ ;
•

$R_{q}$ is from $s_{i}$ to $s_{q}$ ;
•

$R_{p}^{*}\cap R_{q}^{*}=\emptyset$ ; and
•

$R_{p}^{*}\cap\{s_{1},\dots,s_{l}\}=R_{q}^{*}\cap\{s_{1}\dots,s_{l}\}=\emptyset$ .

Now we get a theta with ends $s_{i},a$ and path $s_{i}\hbox{-}Q_{i}\hbox{-}a$ , $s_{i}\hbox{-}R_{p}\hbox{-}s_{p}\hbox{-}Q_{p}\hbox{-}a$ and $s_{i}\hbox{-}R_{q}\hbox{-}s_{q}\hbox{-}Q_{q}\hbox{-}a$ . This proves (6).

Observe that $a_{1}v$ is a $k$ -banana in $G^{\prime}$ . But by (6), $G^{\prime}\in\mathcal{C}_{t}$ , and by (6), $N_{G^{\prime}}(v)\cap\operatorname{Hub}(G^{\prime})=\emptyset$ , contrary to Theorem 3.2. ∎

7. The proof of Theorem 1.3

We can now prove our first main result. The “big picture” of the proof is similar to [3] and [5], but the context here is different. For the remainder of the paper, all logarithms are taken in base 2. We start with the following theorem from [5]:

Theorem 7.1 (Abrishami, Chudnovsky, Hajebi, Spirkl [5]).

Let $t\in\mathbb{N}$ , and let $G$ be (theta, $K_{t}$ )-free with $|V(G)|=n$ . There exist an integer $d=d(t)$ and a partition $(S_{1},\dots,S_{k})$ of $V(G)$ with the following properties:

(1)

$k\leq\frac{d}{4}\log n$ .
(2)

$S_{i}$ is a stable set for every $i\in\{1,\dots,k\}$ .
(3)

For every $i\in\{1,\dots,k\}$ and $v\in S_{i}$ we have $\deg_{G\setminus\bigcup_{j<i}S_{j}}(v)\leq d$ .

Let $G\in\mathcal{C}_{t}$ be a graph and let $a,b\in V(G)$ . A hub-partition with respect to $ab$ of $G$ is a partition $S_{1},\dots,S_{k}$ of $\operatorname{Hub}(G)\setminus\{a,b\}$ as in Theorem 7.1; we call $k$ the order of the partition. We call the hub-dimension of $(G,ab)$ (denoting it by $\operatorname{hdim}(G,ab)$ ) the smallest $k$ such that $G$ has a hub-partition of order $k$ with respect to $ab$ .

For the remainder of this section, let us fix $t\in\mathbb{N}$ . Let $d=d(t)$ be as in Theorem 7.1. Let $m=k+2d$ where $k=k(t)$ is as in Theorem 3.2. Let $C_{t}=(4m+2)(m-1)$ . Let $\Gamma=\max\{k,\Gamma(t)\}$ where $\Gamma(t)$ is as in Theorem 6.3.

Since in view of Theorem 7.1, we have $\operatorname{hdim}(G,ab)\leq\frac{d}{4}\log n$ for all $a,b\in V(G)$ , Theorem 1.3 follows immediately from the next result:

Theorem 7.2.

Let $G\in\mathcal{C}_{t}$ and with $|V(G)|=n$ and let $a,b\in V(G)$ . Then $ab$ is not a $C_{t}+8m^{2}\Gamma\operatorname{hdim}(G,ab)$ -banana in $G$ .

Proof.

Suppose that $ab$ is a $C_{t}+8m^{2}\Gamma\operatorname{hdim}(G,ab)$ -banana in $G$ . We will get a contradiction by induction on $\operatorname{hdim}(G,ab)$ , and for fixed $\operatorname{hdim}$ by induction on $n$ . Suppose that $\operatorname{hdim}(G,ab)=0$ . Then $\operatorname{Hub}(G)\subseteq\{a,b\}$ and by Theorem 3.3, we have that there is no $k$ -banana in $G$ . Now the statement holds since $k<C_{t}$ . Thus we may assume $\operatorname{hdim}(G,ab)>0$ .

(16) We may assume that $G$ admits no clique cutset.

Suppose $C$ is a clique cutset in $G$ . Since $G$ is $K_{t}$ -free, it follows that there is a component $D$ of $G\setminus C$ such that $ab$ is $C_{t}+8m^{2}\Gamma\operatorname{hdim}(G,ab)$ -banana in $G^{\prime}=D\cup C$ . But $|V(G^{\prime})|<n$ and $\operatorname{hdim}(G^{\prime},ab)\leq\operatorname{hdim}(G,ab)$ , consequently we get contradiction (inductively on $n$ ). This proves (7).

Let $S_{1},\dots,S_{q}$ be a hub-partition of $G$ with respect to $ab$ and with $q=\operatorname{hdim}(G,ab)$ . We now use notation and terminology from Section 4 (note that the definitions of $k$ and $m$ agree; we use $d$ , $a$ , $b$ , as defined above). It follows from the definition of $S_{1}$ that every vertex in $S_{1}$ is $d$ -safe. Let $(T,\chi)$ be an $m$ -atomic tree decomposition of $G$ . Let $t_{1},t_{2}$ be as in Theorem 2.6. and let $\beta=\beta(S_{1})$ and $\beta^{A}(S_{1})$ be as in Section 4. Then by Lemma 4.1, we have $a,b\in\beta^{A}(S_{1})$ . By Theorem 4.10(5), we have that $S_{1}\cap\operatorname{Hub}(\beta^{A}(S_{1}))=\emptyset$ and $S_{2}\cap\operatorname{Hub}(\beta^{A}(S_{1})),\dots,S_{q}\cap\operatorname{Hub}(\beta^{A}(S_{1}))$ is a hub-partition of $\beta^{A}(S_{1})$ with respect to $ab$ . It follows that $\operatorname{hdim}(\beta^{A}(S_{1}),ab)\leq q-1$ . Inductively (on $\operatorname{hdim}(\cdot,ab)$ ), we have that $ab$ is not a $C_{t}+8m^{2}\Gamma(q-1)$ -banana in $\beta^{A}(S_{1})$ .

Our first goal is to prove:

(17) There is a set $Y^{\prime}\subseteq\beta$ such that

(1)

$Y^{\prime}$ separates $a$ from $b$ in $\beta$ ;
(2)

$|Y^{\prime}\cap(\chi(t_{1})\cup\chi(t_{2}))|\leq C_{t}+8m^{2}\Gamma(q-1)+4m^{2}\Gamma$ ; and
(3)

$|\Delta(Y^{\prime})|\leq(4m+1)(m-1)\Gamma$ .

Since $ab$ is not a $(C_{t}+8m^{2}\Gamma(q-1))$ -banana in $\beta^{A}(S_{1})$ , Theorem 2.1 implies that there is a separation $(X,Y,Z)$ of $\beta^{A}(S_{1})$ such that $a\in X$ and $b\in Z$ and $|Y|\leq C_{t}+8m^{2}\Gamma(q-1)$ . We may assume that $(X,Y,Z)$ is chosen with $|Y|$ as small as possible. Let $D_{1}$ be the component of $X$ such that $a\in D_{1}$ , and and let $D_{2}$ be the component of $Z$ such that $b\in D_{2}$ . It follows from the minimality of $|Y|$ that $N(D_{1})=N(D_{2})=Y$ and $ab$ is a $|Y|$ -banana in $D_{1}\cup D_{2}\cup Y$ .

We claim that $|Y\cap S_{1}|\leq\Gamma$ . Let $G^{\prime}=D_{1}\cup D_{2}\cup(Y\cap S_{1})$ . Again by the minimality of $Y$ , it follows that $ab$ is a $|Y\cap S_{1}|$ -banana in $G^{\prime}$ . If $ab$ is not a $k$ -banana in $G^{\prime}$ , then $|Y\cap S_{1}|<k$ and the claim follows immediately since $k\leq\Gamma$ ; thus we may assume that $ab$ is a $k$ -banana in $G^{\prime}$ . Now since $S_{1}$ is a stable set and since no vertex of $S_{1}$ is a hub in $\beta^{A}(S_{1})$ , applying Theorem 6.3 to $G^{\prime}$ we deduce that $|Y\cap S_{1}|\leq\Gamma$ , as required. This proves the claim.

Next we claim that $|\delta_{1}(Y)\cup\delta_{2}(Y)|\leq\Gamma$ , and $|\Delta(Y)|\leq(m-1)\Gamma$ . Write $\beta_{0}=\chi(t_{1})\cup\chi(t_{2})$ . Observe that for distinct $t,t^{\prime}\in\delta_{1}(Y)\cup\delta_{2}(Y)$ , say with $t\in\delta_{i}(Y)$ and $t^{\prime}\in\delta_{i^{\prime}}(Y)$ for $i,i^{\prime}\in\{1,2\}$ , the sets $G_{t_{i}\rightarrow t}\setminus\beta_{0}$ and $G_{t_{i^{\prime}}\rightarrow t^{\prime}}\setminus\beta_{0}$ are disjoint and anticomplete to each other. Let $W$ be a subset of $Y$ such that for all $i\in\{1,2\}$ and $t\in\delta_{i}(Y)$ , we have $|W\cap(G_{t_{i}\rightarrow t}\setminus\beta_{0}))|=1$ . Then $W$ is stable set and $|W|=|\delta_{1}(Y)\cup\delta_{2}(Y)|$ . Let $G^{\prime}=D_{1}\cup D_{2}\cup W$ . If follows from the minimality of $Y$ that $ab$ is a $|W|$ -banana in $G^{\prime}$ . If $ab$ is not a $k$ -banana in $G^{\prime}$ , then $|W|<k$ and the claim follows immediately since $k\leq\Gamma$ ; thus we may assume that $ab$ is a $k$ -banana in $G^{\prime}$ . By Theorem 4.4(3), $W\cap\operatorname{Hub}(\beta)=\emptyset$ . Now since $W$ is a stable set, applying Theorem 6.3 to $G^{\prime}$ we deduce that $|W|\leq\Gamma$ , as required. Since $\operatorname{adh}(T,\chi)\leq m-1$ , it follows that $|\Delta(Y)|\leq(m-1)\Gamma$ . This proves the claim.

Now let $Y^{\prime}$ be as in Theorem 4.11. Then $Y^{\prime}$ separates $a$ from $b$ in $\beta$ . Moreover, by Theorem 4.11,

|Y^{\prime}\cap(\chi(t_{1})\cup\chi(t_{2}))|\leq|Y|+4m(m-1)|Y\cap\operatorname{Core}(S_{1})|.

Since $|Y\cap\operatorname{Core}(S_{1})|\leq|Y\cap S_{1}|\leq\Gamma$ , it follows that

|Y^{\prime}\cap(\chi(t_{1})\cup\chi(t_{2}))|\leq C_{t}+8m^{2}\Gamma(q-1)+4m^{2}\Gamma.

Finally, by Theorem 4.11, $|\Delta(Y^{\prime})\setminus\Delta(Y)|\leq 4m(m-1)|Y\cap\operatorname{Core}(S_{1})|$ . Since $|\Delta(Y)|\leq(m-1)\Gamma$ and $|Y\cap\operatorname{Core}(S_{1})|\leq\Gamma$ , it follows that $|\Delta(Y^{\prime})|\leq(4m+1)(m-1)\Gamma$ . This proves (7).

Let $Y^{\prime}$ be as in (7), and let $(X^{\prime},Y^{\prime},Z^{\prime})$ be a separation of $\beta$ such that $a\in X^{\prime}$ and $b\in Z^{\prime}$ . Let $Y^{\prime\prime}$ be obtained from $Y^{\prime}$ as in Theorem 4.12. Then $Y^{\prime\prime}$ separates $a$ from $b$ in $G$ . Recall that by (7), we have $|\Delta(Y^{\prime})|\leq(4m+1)(m-1)\Gamma$ . Now since $\operatorname{adh}(T,\chi)<m$ and since $|{S_{1}}_{bad}|\leq 3$ , it follows from Theorem 4.12 that

	$\displaystyle\|Y^{\prime\prime}\|$	$\displaystyle\leq\|Y^{\prime}\cap(\chi(t_{1})\cup\chi(t_{2}))\|+(4m+1)(m-1)\Gamma+2(m-1)+3$
		$\displaystyle\leq\|Y^{\prime}\cap(\chi(t_{1})\cup\chi(t_{2}))\|+4m^{2}\Gamma.$

Since $|Y^{\prime}\cap(\chi(t_{1})\cup\chi(t_{2}))|\leq C_{t}+8m^{2}\Gamma(q-1)+4m^{2}\Gamma$ by (7), we deduce that $|Y^{\prime\prime}|\leq C_{t}+8m^{2}\Gamma q$ as required. ∎

8. Dominated balanced separators

Several more steps are needed to complete the proof of Theorem 1.5. We take the first one in this section. The goal of this section is to prove Theorem 1.4, which we restate:

Theorem 8.1.

There is an integer $d$ with the following property. Let $G\in\mathcal{C}$ and let $w$ be a normal weight function on $G$ . Then there exists $Y\subseteq V(G)$ such that

•

$|Y|\leq d$ , and
•

$N[Y]$ is a $w$ -balanced separator in $G$ .

We start with two decomposition theorems that will become the engine for obtaining separators. The first is a more explicit version of Theorem 3.1; it shows that the presence of a wheel in the graph forces a decomposition that is an extension of what can be locally observed in the wheel (see [4] for detailed treatment of this concept).

Theorem 8.2 (Addario-Berry, Chudnovsky, Havet, Reed, Seymour [7], da Silva, Vušković [17]).

Let $G$ be a $C_{4}$ -free odd-signable graph that contains a proper wheel $(H,x)$ that is not a universal wheel. Let $x_{1}$ and $x_{2}$ be the endpoints of a long sector $Q$ of $(H,x)$ . Let $W$ be the set of all vertices $h$ in $H\cap N(x)$ such that the subpath of $H\setminus\{x_{1}\}$ from $x_{2}$ to $h$ contains an even number of neighbors of $x$ , and let $Z=H\setminus(Q\cup N(x))$ . Let $N^{\prime}=N(x)\setminus W$ . Then, $N^{\prime}\cup\{x\}$ is a cutset of $G$ that separates $Q^{*}$ from $W\cup Z$ .

The second is a similar kind of theorem, but we start with a pyramid rather than a wheel.

Theorem 8.3.

Let $G\in\mathcal{C}$ . Let $\Sigma$ be a pyramid with paths $P_{1},P_{2},P_{3}$ , apex $a$ , and base $b_{1}b_{2}b_{3}$ in $G$ and let $i,j\in\{1,2,3\}$ be distinct. For $i\in\{1,\dots,3\}$ , let $a_{i}$ be the neighbor of $a$ in $P_{i}$ . Let $P$ be a path from a vertex of $P_{i}\setminus\{a,a_{i},b_{i}\}$ to a vertex of $P_{j}\setminus\{a,a_{j},b_{j}\}$ . Then at least one one of $a,b_{1},b_{2},b_{3}$ has a neighbor in $P^{*}$ .

Proof.

It follows from the defintion of $P$ that there exist distinct $i,j\in\{1,2,3\}$ and a subpath $P^{\prime}=p_{1}\hbox{-}\cdots p_{k}$ of $P$ such that $p_{1}$ has a neighbor in $P_{i}\setminus\{a,a_{i},b_{i}\}$ , and $p_{k}$ has a neighbor in $P_{j}\setminus\{a,a_{j},b_{j}\}$ . Let $P^{\prime}$ be chosen minimal with this property. It follows that $P^{\prime}\subseteq P^{*}$ , and so we may assume that $\{a,b_{1},b_{2},b_{3}\}$ is anticomplete to $P^{\prime}$ . We may also assume $i=1$ and $j=3$ . See Figure 7. It follows that and $a_{1}\neq b_{1}$ , $a_{3}\neq b_{3}$ .

(18) $P^{\prime}$ is anticomplete to $P_{2}\setminus a_{2}$ .

Suppose not; then some vertex in $P^{\prime}$ has a neighbor in $P_{2}\setminus\{a,a_{2},b_{2}\}$ . It follows from the minimality of $P^{\prime}$ that $k=1$ . Then $p_{1}$ has a neighbor in each of the paths $P_{1}\setminus b_{1},P_{2}\setminus b_{2}$ and $P_{3}\setminus b_{3}$ . Now we get a theta with ends $p_{1},a$ whose paths are subpaths of $P_{1},P_{2},P_{3}$ , a contradiction. This proves (8).

(19) If $k>1$ and $a_{1}$ is adjacent to $p_{k}$ , then $a_{1}$ has a neighbor in $P^{\prime}\setminus p_{k}$ .

Suppose that $a_{1}$ is adjacent to $p_{k}$ and anticomplete to $P^{\prime}\setminus p_{k}$ . Since $a_{2}\hbox{-}a\hbox{-}a_{1}\hbox{-}p_{k}\hbox{-}a_{2}$ is not a $C_{4}$ in $G$ , and by (8), it follows that $p_{k}$ is anticomplete to $P_{2}$ . If $a_{2}$ has a neighbor in $P^{\prime}\setminus p_{1}$ , then we get a theta with ends $a_{1},a_{2}$ , two of whose paths are contained in the hole $b_{1}\hbox{-}P_{1}\hbox{-}a\hbox{-}P_{2}\hbox{-}b_{2}\hbox{-}b_{1}$ , and third one has interior on $P^{\prime}\setminus p_{1}$ , a contradiction. Consequently, $a_{2}$ is anticomplete to $P^{\prime}\setminus p_{1}$ . Next suppose that $a_{3}$ has a neighbor in $P^{\prime}$ . Since $a_{1}\hbox{-}a\hbox{-}a_{3}\hbox{-}p_{k}\hbox{-}a_{1}$ is not a $C_{4}$ in $G$ , it follows that $a_{3}$ is not adjacent to $p_{k}$ . Now we get a theta with ends $a_{3},p_{k}$ and path $p_{k}\hbox{-}a_{1}\hbox{-}a\hbox{-}a_{3}$ , a path with interior in $P^{\prime}$ and a path with interior in $P_{3}$ . This proves that $a_{3}$ is anticomplete to $P^{\prime}$ .

Let $x$ be the neighbor of $a_{1}$ in $P_{1}\setminus a$ . Suppose that $p_{1}$ has a neighbor in $P_{1}\setminus\{a,a_{1},x\}$ . Then there is a path $T$ from $p_{1}$ to $a$ with $T^{*}\subseteq(P_{1}\cup P_{2})\setminus\{a,a_{1},x\}$ and we get a theta with ends $p_{k},a$ and paths $p_{k}\hbox{-}a_{1}\hbox{-}a$ and $p_{k}\hbox{-}P^{\prime}\hbox{-}p_{1}\hbox{-}T\hbox{-}a$ , and a path with interior in $P_{3}^{*}$ , a contradiction. It follows that $p_{1}$ is anticomplete to $P_{1}\setminus\{a_{1},a,x\}$ and therefore $p_{1}$ is adjacent to $x$ . Since $a_{1}$ is anticomplete to $P^{\prime}\setminus p_{k}$ we deduce that $N_{P_{1}}(p_{1})=\{x\}$ . Now we get a theta with ends $x,p_{k}$ and paths $x\hbox{-}a_{1}\hbox{-}p_{k}$ , $x\hbox{-}P^{\prime}\hbox{-}p_{k}$ and $x\hbox{-}P_{1}\hbox{-}b_{1}\hbox{-}b_{3}\hbox{-}P_{3}\hbox{-}p_{k}$ , a contradiction. This proves (8).

(20) Either $N_{P^{\prime}}(a_{1})\subseteq\{p_{1}\}$ or $N_{P^{\prime}}(a_{3})\subseteq\{p_{k}\}$ .

Suppose not. Then $k>1$ . If both $a_{1}$ and $a_{3}$ have a neighbor in $P^{\prime*}$ , then there is a theta in $G$ with ends $a_{1},a_{3}$ , and paths $a_{1}\hbox{-}a\hbox{-}a_{3},a_{1}\hbox{-}P_{1}\hbox{-}b_{1}\hbox{-}b_{3}\hbox{-}P_{3}\hbox{-}a_{3}$ and $a_{1}\hbox{-}P^{\prime}\hbox{-}a_{3}$ , a contradiction. By symmetry we may assume that $a_{1}$ is anticomplete to $P^{\prime*}$ ; now by (8), it follows that $a_{1}$ is adjacent to both $p_{1}$ and $p_{k}$ . Since $a_{1}\hbox{-}p_{1}\hbox{-}a_{3}\hbox{-}a\hbox{-}a_{1}$ is not a $C_{4}$ , we deduce that $a_{3}$ is non-adjacent to $p_{1}$ . Similarly $a_{3}$ is non-adjacent to $p_{k}$ . It follows that $a_{3}$ has a neighbor in $P^{\prime*}$ . Let $S$ be a path from $a_{3}$ to $p_{k}$ with interior in $P^{\prime*}$ . Now we get a theta with ends $a_{3},p_{k}$ and path $a_{3}\hbox{-}S\hbox{-}p_{k}$ , $a_{3}\hbox{-}a\hbox{-}a_{1}\hbox{-}p_{k}$ and a path with interior in $P_{3}$ , a contradiction. This proves (8).

(21) If $a_{1}$ has a neighbor in $P^{\prime}\setminus p_{1}$ , then $a_{2}$ is anticomplete to $P^{\prime}$ .

Suppose that $a_{1}$ has a neighbor in $P^{\prime}\setminus p_{1}$ and $a_{2}$ has a neighbor in $P^{\prime}$ . Then $k>1$ . If both $a_{1}$ and $a_{2}$ have a neighbor in $P^{\prime}\setminus p_{1}$ , then there is a theta in $G$ with ends $a_{1},a_{2}$ , and paths $a_{1}\hbox{-}a\hbox{-}a_{2},a_{1}\hbox{-}P_{1}\hbox{-}b_{1}\hbox{-}b_{2}\hbox{-}P_{2}\hbox{-}a_{2}$ and $a_{1}\hbox{-}P^{\prime}\hbox{-}a_{2}$ , a contradiction. This proves that $N_{P^{\prime}}(a_{2})=\{p_{1}\}$ .

Let $S_{1}$ be a path from $a_{1}$ to $p_{1}$ with $S_{1}^{*}\subseteq P^{\prime}$ . Since $a_{1}\hbox{-}p_{1}\hbox{-}a_{2}\hbox{-}a\hbox{-}a_{1}$ is not a $C_{4}$ , it follows that $a_{1}$ is non-adjacent to $p_{1}$ . But now we get a theta with ends $a_{1},p_{1}$ and paths $a_{1}\hbox{-}a\hbox{-}a_{2}\hbox{-}p_{1}$ , $a_{1}\hbox{-}S_{1}\hbox{-}p_{1}$ and a path from $a_{1}$ to $p_{1}$ with interior in $P_{1}$ , a contradiction. This proves (8).

(22) $N_{P^{\prime}}(a_{1})\subseteq\{p_{1}\}$ and $N_{P^{\prime}}(a_{3})\subseteq\{p_{k}\}$ .

Suppose $a_{1}$ has a neighbor in $P^{\prime}\setminus p_{1}$ . Then $k>1$ . By (8), $a_{3}$ is anticomplete to $P^{\prime}\setminus p_{k}$ , and by (8), $a_{2}$ is anticomplete to $P^{\prime}$ . Let $H_{1}$ be the hole $b_{1}\hbox{-}P_{1}\hbox{-}p_{1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}P_{3}\hbox{-}b_{3}\hbox{-}b_{1}$ , and let $H_{2}$ be the hole $p_{1}\hbox{-}P_{1}\hbox{-}b_{1}\hbox{-}b_{2}\hbox{-}P_{2}\hbox{-}a\hbox{-}P_{3}\hbox{-}p_{k}\hbox{-}P^{\prime}\hbox{-}p_{1}$ ( $H_{2}$ is a hole since $p_{k}$ has a neighbor in $P_{3}\setminus b_{3}$ ). Then $N_{H_{2}}(a_{1})=N_{H_{1}}(a_{1})\cup\{a\}$ . Let $p_{0}$ be the neighbor of $p_{1}\in H_{1}\setminus P^{\prime}$ and let $Q$ be the path $p_{0}\hbox{-}p_{1}\hbox{-}P^{\prime}\hbox{-}p_{k}$ . Observe that $Q\subseteq H_{2}$ , and $N_{H_{1}}(a_{1})\subseteq Q$ . Since neither of $(H_{1},a_{1})$ and $(H_{2},a_{1})$ is an even wheel, it follows that $a_{1}$ has exactly two neighbors in $Q$ and they are adjacent. Let $s\in\{0,\dots,k-1\}$ be such that $N_{Q}(a_{1})=\{p_{s},p_{s+1}\}$ . Since $a_{1}$ has a neighbor in $P_{1}\setminus p_{1}$ , it follows that $s>0$ . Now we get a prism with triangles $p_{s}a_{1}p_{s+1}$ and $b_{1}b_{2}b_{3}$ and paths $a_{1}\hbox{-}a\hbox{-}P_{2}\hbox{-}b_{2}$ , $p_{s}\hbox{-}P^{\prime}\hbox{-}p_{1}\hbox{-}p_{0}\hbox{-}P_{1}\hbox{-}b_{1}$ and $p_{s+1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}P_{3}\hbox{-}b_{3}$ , a contradiction. This proves (8).

(23) $P_{2}$ is anticomplete to $P^{\prime}$ .

Suppose not. By (8), $a_{2}$ has a neighbor in $P^{\prime}$ . Then $a_{2}\neq b_{2}$ . Let $H_{1}$ be the hole $p_{1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}P_{3}\hbox{-}b_{3}\hbox{-}b_{1}\hbox{-}P_{1}\hbox{-}p_{1}$ and let $H_{2}$ be the hole $p_{1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}P_{3}\hbox{-}a_{3}\hbox{-}a\hbox{-}a_{1}\hbox{-}P_{1}\hbox{-}p_{1}$ ( $H_{2}$ is a hole since $\{b_{1},b_{3}\}$ is anticomplete to $P^{\prime}$ ). Since $a_{2}\neq b_{2}$ , we have that $N_{H_{2}}(a_{2})=N_{H_{1}}(a_{2})\cup\{a\}$ . Since since neither of $(H_{1},a_{2})$ , $(H_{2},a_{2})$ is an even wheel, it follows that there exists $s\in\{1,\dots,k-1\}$ such that $N_{P^{\prime}}(a_{2})=\{p_{s},p_{s+1}\}$ . Now we get a prism with triangles $b_{1}b_{2}b_{3}$ and $p_{s}a_{2}p_{s+1}$ and paths $p_{s}\hbox{-}P^{\prime}\hbox{-}p_{1}\hbox{-}P_{1}\hbox{-}b_{1}$ , $a_{2}\hbox{-}P_{2}\hbox{-}b_{2}$ and $p_{s+1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}P_{3}\hbox{-}b_{3}$ , a contradiction. This proves (8).

(24) $p_{1}$ has at least two neighbors in $P_{1}$ , and $p_{3}$ has at least two neighbors in $P_{3}$ .

By symmetry it is enough to prove the first assertion of (8). Suppose that $p_{1}$ has a unique neighbor $x$ in $P_{1}$ . It follows that $x\neq a_{1}$ . By (8), $P_{2}$ is anticomplete to $P^{\prime}$ . Now we get a theta with ends $x,a$ and paths $x\hbox{-}P_{1}\hbox{-}a$ , $x\hbox{-}P_{1}\hbox{-}b_{1}\hbox{-}b_{2}\hbox{-}P_{2}\hbox{-}a$ and $x\hbox{-}p_{1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}P_{3}\hbox{-}a$ , a contradiction. This proves (8).

(25) $p_{1}$ has two non-adjacent neighbors in $P_{1}$ , and $p_{3}$ has two non-adjacent two neighbors in $P_{3}$ .

By symmetry it is enough to prove the first statement. By (8), we may assume that $p_{1}$ has exactly two neighbors $x,y$ in $P_{1}$ , and $x$ is adjacent to $y$ . We may assume that $P_{1}$ traverses $b_{1},x,y,a_{1}$ in this order. By (8), $P_{2}$ is anticomplete to $P^{\prime}$ . Let $S$ be the path from $p_{k}$ to $b_{3}$ with interior in $P_{3}$ . It follows from the defintion of $P^{\prime}$ that $a$ is anticomplete to $S$ . Now we get a prism with triangles $xyp_{1}$ and $b_{1}b_{2}b_{3}$ and paths $x\hbox{-}P_{1}\hbox{-}b_{1}$ , $y\hbox{-}P_{1}\hbox{-}a\hbox{-}P_{2}\hbox{-}b_{2}$ and $p_{1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}S\hbox{-}b_{3}$ , a contradiction. This proves (8).

By (8), there exist paths $P_{1}^{\prime}$ from $p_{1}$ to $a$ and $P_{1}^{\prime\prime}$ from $p_{1}$ to $b_{1}$ , both with interior in $P_{1}^{*}$ , and such that $P_{1}^{\prime}\setminus p_{1}$ is anticomplete to $P_{1}^{\prime\prime}\setminus p_{1}$ . Let $P_{3}^{\prime}$ be a path from $p_{k}$ to $a$ with interior in $P_{3}^{*}$ . By (8), $P_{2}$ is anticomplete to $P^{\prime}$ . Now we get a theta with ends $p_{1},a$ and paths $p_{1}\hbox{-}P_{1}^{\prime}\hbox{-}a$ , $p_{1}\hbox{-}P_{1}^{\prime\prime}\hbox{-}b_{1}\hbox{-}b_{2}\hbox{-}P_{2}\hbox{-}a$ and $p_{1}\hbox{-}P^{\prime}\hbox{-}p_{k}\hbox{-}P_{3}^{\prime}\hbox{-}a$ , a contradiction. ∎

We need a technical definition. Given an integer $m>0$ and a graph class $\mathcal{G}$ , we say $\mathcal{G}$ is $m$ -amicable if $\mathcal{G}$ is amiable, and the following holds for every graph $G\in\mathcal{G}$ . Let $\sigma$ be as in the definition of an amiable class for $\mathcal{G}$ and let $V(G)=D_{1}\cup D_{2}\cup Y$ such that $D_{1}=d_{1}\hbox{-}\cdots\hbox{-}d_{k},D_{2}$ and $Y$ satisfy the first five bullets from the definition of an amiable class with $|Y|=\sigma(7)$ . Let $X\subseteq Y$ , $H_{1}\subseteq D_{1}$ and $H_{2}\subseteq D_{2}$ be as in the definition of an amiable class with $|X|=9$ , and let $\{x_{1},\dots,x_{7}\}\subseteq X$ such that:

•

$x_{1},\dots,x_{7}$ is the order given on $\{x_{1},\dots,x_{7}\}$ by $(H_{1},X)$ ; and
•

For every $x\in\{x_{1},\dots,x_{7}\}$ , we have $N_{D_{1}}(x)=N_{H_{1}}(x)$ .

Let $i$ be maximum such that $x_{1}$ is adjacent to $d_{i}$ , and let $j$ be minimum such that $x_{7}$ is adjacent to $d_{j}$ . Then there exists a subset $Z\subseteq D_{2}\cup\{d_{i+2},\dots,d_{j-2}\}\cup\{x_{4}\}$ with $|Z|\leq m$ such that $N[Z]$ separates $d_{i}$ from $d_{j}$ . It follows that $N[Z]$ separates $d_{1}\hbox{-}D_{1}\hbox{-}d_{i}$ from $d_{j}\hbox{-}D_{1}\hbox{-}d_{k}$ .

We deduce:

Theorem 8.4.

The class $\mathcal{G}$ is $4$ -amicable.

Proof.

From Theorem 5.7, we know that $\mathcal{G}$ is amiable. With the notation as in the definition of an amicable class, we need to show that there exists a subset $Y\subseteq D_{2}\cup\{d_{i+2},\dots,d_{j-2}\}\cup\{x_{4}\}$ with $|Z|\leq 4$ such that $N[Z]$ separates $d_{i}$ from $d_{j}$ . Let $R$ be the path from $x_{1}$ to $x_{7}$ with interior in $H_{2}$ .

Let $H$ be the hole $x_{1}\hbox{-}H_{1}\hbox{-}x_{7}\hbox{-}R\hbox{-}x_{1}$ . Let $L=l_{1}\hbox{-}\cdots\hbox{-}l_{q}$ be the path in $H_{2}\cup\{x_{4}\}$ such that $l_{1}=x_{4}$ and $l_{q}$ has a neighbor in $P(H_{2})$ . Thus if $(H_{2},X)$ is an alignment then $L=x_{4}$ , and if $(H_{2},X)$ is a connectifier then $L\setminus x_{4}$ is the leg of $H_{2}$ containing a neighbor of $x_{4}$ . See Figure 8. (Note that $L\setminus x_{4}$ may be empty if $(H_{2},X)$ is a clique connectifier or a stellar connectifier.) We consider different possibilities for the behavior of $(H_{1},X)$ and $(H_{2},X)$ .

(26) If $(H_{1},X)$ is a wide alignment, then the theorem holds.

Suppose first that $(H_{2},X)$ is an alignment. Then $(H,x_{4})$ is a proper wheel, and we are done by Theorem 8.2 and setting $Z=\{x_{4}\}$ . Thus we may assume that $(H_{2},X)$ is a connectifier. It now follows from Theorem 5.7 that $(H_{2},X)$ is a triangular connectifier or a clique connectifier. Let $N_{H}(l_{q})=\{h,h^{\prime}\}$ . Then $H\cup L$ contains a pyramid $\Sigma$ with apex $x_{4}$ and base $l_{q}hh^{\prime}$ . Since $\{d_{i},d_{j}\}$ is anticomplete to $\{x_{4},h,h^{\prime},l_{q}\}$ , it follows that $d_{i}$ belongs to the interior of the path of $\Sigma$ with ends $x_{4},h$ , and $d_{j}$ belongs to the interior of the path of $\Sigma$ with ends $x_{4},h^{\prime}$ (switching the roles of $h,h^{\prime}$ if necessary), and that $d_{i},d_{j}$ are not adjacent to the apex $x_{4}$ of $\Sigma$ . Since $\{d_{i},d_{j}\}\cap N(x_{4})=\emptyset$ , Theorem 8.3 implies that $N[x_{4},h,h^{\prime},l_{1}]$ separates $d_{i}$ from $d_{j}$ . Since $h,h^{\prime},l_{q}\in D_{2}$ , we are done. This proves (8).

(27) If $(H_{2},X)$ is a wide alignment, then the theorem holds.

Since $(H_{1},X)$ is an alignment, we deduce that $(H,x_{4})$ is a proper wheel. Now by Theorem 8.2 and setting $Z=\{x_{4}\}$ , we are done. This proves (8).

(28) If $(H_{1},X)$ is a triangular alignment, then the theorem holds.

Let $l\in\{i,\dots,j\}$ be such that $N(x_{4})\cap D_{1}=\{d_{l},d_{l+1}\}$ . Since $x_{2}$ has a neighbor in the path $d_{i}\hbox{-}H_{1}\hbox{-}d_{l}$ , it follows that $l>i+1$ . Since $x_{6}$ has a neighbor in the path $d_{l+1}\hbox{-}H_{1}\hbox{-}d_{j}$ , it follows that $l<j-2$ . By Theorem 5.7 and in view of (8), we may assume that $(H_{2},X)$ is a spiky alignment, a stellar connectifier or a spiky connectifier. In all cases, $l_{q}$ has a unique neighbor in $P(H_{2})$ ; denote it by $s$ (where $P(H_{2})$ is as defined in Section 5). Now $H\cup L$ is a pyramid $\Sigma$ with apex $s$ and base $x_{4}d_{l}d_{l+1}$ . Since $s\in D_{2}$ and since $x_{4}$ is non-adjacent to $d_{i},d_{j}$ , it follows that $d_{i}$ belongs to the interior of the path of $\Sigma$ with ends $d_{l},s$ , and $d_{j}$ belongs to the interior of the path of $\Sigma$ with ends $d_{l+1},s$ , and $d_{i},d_{j}$ are not adjacent to the apex $s$ of $\Sigma$ . Since $\{d_{i},d_{j}\}\cap N(s)=\emptyset$ , Theorem 8.3 implies that $N[x_{4},s,d_{l},d_{l+1}]$ separates $d_{i}$ from $d_{j}$ . Since $s\in D_{2}$ and $i+1<l<j-2$ , we are done. This proves (8).

By Theorem 5.7 and in view of (8) and (8), we deduce that $(H_{1},X)$ is a spiky alignment. Let $d_{l}$ be the unique neighbor of $x_{4}$ in $H_{1}$ . By Theorem 5.7 and in view of (8), it follows that $(H_{2},X)$ is either a triangular alignment or a triangular connectifier or a clique connectifier. Let $h,h^{\prime}$ be the neighbors of $l_{q}$ in $H$ . Now $H\cup L$ is a pyramid $\Sigma$ with apex $d_{l}$ and base $l_{q}hh^{\prime}$ . Since $x_{2}$ has a neighbor in the path $d_{i}\hbox{-}H_{1}\hbox{-}d_{l}$ , it follows from the definition of a spiky alignment that $i<l-1$ . Similarly, $j>l+1$ . Since $h,h^{\prime}\in D_{2}$ , we deduce that $d_{i}$ belongs to the interior of the path of $\Sigma$ with ends $d_{l},h$ , and $d_{j}$ belongs to the interior of the path of $\Sigma$ with ends $d_{l},h^{\prime}$ (possibly switching the roles of $h,h^{\prime}$ ), and $d_{i},d_{j}$ are non-adjacent to the apex $d_{l}$ of $\Sigma$ since $i+1<l<j-1$ . Therefore, Theorem 8.3 implies that $N[d_{l},h,h^{\prime},l_{q}]$ separates $d_{i}$ from $d_{j}$ . Since $h,h^{\prime}\in D_{2}$ and $i+1<l<j-1$ , we conclude that $\mathcal{C}$ is 4-amicable.

We now prove that $N[Z]$ separates $d_{1}\hbox{-}D_{1}\hbox{-}d_{i}$ from $d_{j}\hbox{-}D_{1}\hbox{-}d_{k}$ . Let $D$ be the component of $G\setminus N[Z]$ such that $d_{i}\in D$ , and let $D^{\prime}$ be the component of $G\setminus N[Z]$ such that $d_{j}\in D^{\prime}$ . Then $D\neq D^{\prime}$ . Since $Y\subseteq D_{2}\cup\{d_{i+2},\dots,d_{j-2}\}\cup\{x_{4}\}$ , it follows that $N[Z]$ is disjoint from $d_{1}\hbox{-}D_{1}\hbox{-}d_{j}$ and $d_{j}\hbox{-}D_{1}\hbox{-}d_{k}$ . Therefore, $d_{1}\hbox{-}D_{1}\hbox{-}d_{j}\in D$ and $d_{j}\hbox{-}D_{1}\hbox{-}d_{k}\in D^{\prime}$ as required. ∎

We need the following lemma:

Lemma 8.5 (Chudnovsky, Pilipczuk, Pilipczuk, Thomassé [14], Lemma 5.3).

Let $G$ be a connected graph with a weight function $w$ . Then there is an induced path $P=p_{1}\hbox{-}\dots\hbox{-}p_{k}$ in $G$ such that $N[P]$ is a $w$ -balanced separator.

For a graph $G$ and a set $X\subseteq G$ we denote by $\gamma(X)$ the minimum size of a set $Y$ such that $X\subseteq N[Y]$ . We are now ready to prove Theorem 8.1. We prove the following strengthening, which along with Theorem 8.4 yields Theorem 8.1 at once. This will be used in a future paper.

Theorem 8.6.

For every integer $m>0$ and every $m$ -amicable graph class $\mathcal{G}$ , there is an integer $d>0$ with the following property. Let $G\in\mathcal{G}$ and let $w$ be a normal weight function on $G$ . Then there exists $Y\subseteq V(G)$ such that

•

$|Y|\leq d$ , and
•

$N[Y]$ is a $w$ -balanced separator in $G$ .

Proof.

It suffices to consider the unique component of $G$ with weight greater than $1/2$ ; if there is no such component, we set $Y=\emptyset$ . Therefore, we may assume that $G$ is connected.

Since $\mathcal{G}$ is amicable, it is also amiable. Let $K=\sigma(7)$ where $\sigma$ is as in the definition of an amiable class for $\mathcal{C}$ . Let $d=6K+2m+1$ . We claim that $d$ satisfies the conclusion of the theorem.

By Lemma 8.5, there is a path $P$ in $G$ such that $w(D)\leq\frac{1}{2}$ for every component $D$ of $G\setminus N[P]$ . Let $P=p_{1}\hbox{-}\cdots\hbox{-}p_{k}$ be such a path chosen with $k$ as small as possible. It follows from the minimality of $k$ that there is a component $B$ of $G\setminus N[p_{1}\hbox{-}P\hbox{-}p_{k-1}]$ with $w(B)>\frac{1}{2}$ . Let $T=N(P\setminus p_{k})\cap N(B)$ .

(29) Let $D$ be a connected subset of $G$ such that $D\cap(T\cup N[p_{k}])=\emptyset$ . Then $w(D)\leq\frac{1}{2}$ .

Since $D\cap T=\emptyset$ and $D$ is connected, it follows that either $D\subseteq B$ or $D\cap B=\emptyset$ . If $D\cap B=\emptyset$ , then $w(D)\leq 1-w(B)<\frac{1}{2}$ . Thus we may assume that $D\subseteq B$ . Since $D\cap N[p_{k}]=\emptyset$ , we deduce that $D$ is contained in a component of $G\setminus N[P]$ , and so $w(D)\leq\frac{1}{2}$ . This proves (8).

Let $r$ be minimum such that $\gamma[T\cap N(p_{r+1},\dots,p_{k-1})]\leq 2K$ .

(30) We may assume that $r>0$ .

Suppose $r=0$ . Then $\gamma(T)<2K$ . Let $Y^{\prime}\subseteq V(G)$ be such that $T\subseteq N[Y^{\prime}]$ and with $|Y^{\prime}|\leq 2K$ . Let $Y=Y^{\prime}\cup\{p_{k}\}$ . Then every component of $G\setminus N[Y]$ is disjoint from $T\cup N[p_{k}]$ and (8) implies that $Y$ satisfies the conclusion of the theorem. This proves (8).

Let $Y_{0}^{\prime}\subseteq V(G)$ be such that $T\cap N(p_{r},\dots,p_{k-1})\subseteq N[Y_{0}^{\prime}]$ and with $|Y_{0}^{\prime}|\leq 2K$ . Let $Y_{0}=Y_{0}^{\prime}\cup\{p_{k}\}$ . For every $i\in\{1,\dots,r\}$ , let $\operatorname{end}(i)$ be the minimum $j>i$ such that $\gamma(T\cap N[\{p_{i},\dots,p_{\operatorname{end}(i)}\}])=2K$ . Let $Z_{i}$ be a set of size at most $2K$ such that $T\cap N[\{p_{i},\dots,p_{\operatorname{end}(i)}\}]\subseteq N[Z_{i}]$ .

Our next goal is, for every $i\in\{1,\dots,r\}$ , to define two sets $Q_{i}$ and $N_{i}$ . To that end, we fix $i\in\{1,\dots,r\}$ . Let $T_{1}^{\prime}=T\cap N(\{p_{i},\dots,p_{\operatorname{end}(i)}\})$ . Let $S_{1}^{\prime}\subseteq\{p_{i},\dots,p_{\operatorname{end}(i)}\}$ be such that $T_{1}^{\prime}\subseteq N(S_{1}^{\prime})$ , and assume that $S_{1}^{\prime}$ is chosen with $|S_{1}^{\prime}|$ as small as possible. Order the vertices of $S_{1}^{\prime}$ as $q_{1},\dots,q_{l}$ in the order that they appear in $P$ when $P$ is traversed starting from $p_{1}$ . It follows from the minimality of $|S_{1}^{\prime}|$ that there is $n_{1}\in T_{1}^{\prime}$ such that $N_{S_{1}^{\prime}}(n_{1})=\{q_{1}\}$ . Let $t(1)=1$ and let $S_{1}=\{q_{t(1)}\}$ and $T_{1}=\{n_{1}\}$ . So far we have defined $S_{1}^{\prime},S_{1},T_{1}^{\prime},T_{1}$ and $t(1)$ . Suppose that we have defined sequences of subsets $S_{1}^{\prime},\dots,S_{s}^{\prime}$ , $S_{1},\dots,S_{s}$ , $T_{1}^{\prime},\dots,T_{s}^{\prime}$ , and $T_{1},\dots,T_{s}$ and a sequence of integers $t(1),\dots,t(s)$ such that $T_{s}^{\prime}\subseteq N(S_{s}^{\prime})$ and with the following properties (for every $l\in\{1,\dots,s\}$ ):

•

$T_{l}$ is a stable set.
•

If $t<t(l)$ and $q_{t}\in S_{l}^{\prime}$ , then $q_{t}$ is anticomplete to $T_{l}^{\prime}$ .
•

$t(j)<t(l)$ for every $j<l$ .
•

If $j<l$ , then $n_{j}$ is anticomplete to $S_{l}^{\prime}$ , and $N_{S_{l}^{\prime}}(n_{l})=q_{t(l)}$ .
•

For every $j\in\{1,\dots,l\}$ , $N_{S_{l}}(n_{j})=q_{t(j)}$ .

See Figure 9. Now we either terminate the construction or construct the sets $T_{s+1}^{\prime}$ , $T_{s+1}$ , $S_{s+1}^{\prime}$ , $S_{s+1}$ and define $t(s+1)$ , and verify that the properties above continue to hold. Let $T_{s+1}^{\prime}=T_{s}^{\prime}\setminus N[S_{s}\cup T_{s}]$ . If $T_{s+1}^{\prime}=\emptyset$ , let $Q_{i}=S_{s}$ and $N_{i}=T_{s}$ ; the construction terminates here. Now suppose that $T_{s+1}^{\prime}\neq\emptyset$ . Let $S_{s+1}^{\prime}$ be a minimal subset of $S_{s}^{\prime}$ such that $T_{s+1}^{\prime}\subseteq N(S_{s+1}^{\prime})$ . It follows from the second bullet and the minimality of $t(s)$ that if $q_{t}\in S_{s}^{\prime}$ and $t<t(s)$ , then $q_{t}$ is anticomplete to $T_{s}^{\prime}$ . Since $S_{s+1}^{\prime}\subseteq S_{s}^{\prime}$ and $T_{s+1}^{\prime}\subseteq T_{s}^{\prime}$ , we deduce that if $q_{t}\in S_{s+1}^{\prime}$ , then $t>t(s)$ . Let $t$ be minimum such that $q_{t}\in S_{s+1}^{\prime}$ and set $t(s+1)=t$ . Then $t(s+1)>t(s)$ , and the third bullet continues to hold. By the minimality of $t$ , the second bullet continues to hold. It follows from the minimality of $S_{s+1}^{\prime}$ that there exists $n_{s+1}\in T_{s+1}^{\prime}$ such that $N_{S_{s+1}^{\prime}}(n_{s+1})=\{q_{t(s+1)}\}$ . Set $S_{s+1}=S_{s}\cup\{q_{t(s+1)}\}$ and $T_{s+1}=T_{s}\cup\{n_{s+1}\}$ . Since $T_{s}$ is anticomplete to $T_{s+1}^{\prime}$ , the first bullet continues to hold. Since $N_{S_{s+1}^{\prime}}(n_{s+1})=q_{t(s+1)}$ , and since $S_{s+1}^{\prime}\subseteq S_{s}^{\prime}\setminus\{q_{t(s)}\}$ , it follows that the fourth bullet continues to hold, and consequently the fifth bullet continues to hold. Thus our construction maintains the required properties. When we finish, write $N_{i}=\{n_{1},\dots,n_{m}\}$ and $Q_{i}=\{q_{t(1)},\dots,q_{t(m)}\}$ , and we have:

(31) The following statements hold.

•

$N_{i}$ is a stable set.
•

For every $j\in\{1,\dots,m\}$ , $N_{Q_{i}}(n_{j})=q_{t(j)}$ .
•

If $j<j^{\prime}$ , then $t(j)<t(j^{\prime})$ .
•

$T\cap N(\{p_{i},\dots,p_{\operatorname{end}(i)}\})\subseteq N[Q_{i}\cup N_{i}]$ .

To simplify notation, we renumber the elements of $Q_{i}$ , replacing each index $t(j)$ by $j$ . In the new notation we have $Q_{i}=\{q_{1},\dots,q_{m}\}$ , and for every $j\in\{1,\dots,m\}$ , $N_{Q}(n_{j})=\{q_{j}\}$ . Observe that $q_{1},\dots,q_{m}$ appear in this order in $P$ when $P$ is traversed from $p_{1}$ to $p_{k}$ . Let $L_{i}=p_{1}\hbox{-}P\hbox{-}q_{1}$ , $M_{i}=q_{1}\hbox{-}P\hbox{-}q_{m}$ and $R_{i}=q_{m+1}\hbox{-}P\hbox{-}p_{k}$ .

(32) For every $i$ , there exists $r_{i}\in N_{i}$ and $Y_{i}\subseteq(G\setminus N[P])\cup M_{i}\cup\{r_{i}\}$ with $|Y_{i}|\leq m$ such that $N[Y_{i}]$ separates $L_{i}$ from $R_{i}$ .

By (8), $N_{i}$ is a stable set. Since $T\cap N(\{p_{i},\dots,p_{\operatorname{end}(i)}\})\subseteq N[Q_{i}\cup N_{i}]$ , and $|Q_{i}|=|N_{i}|$ , it follows from (8) that $|N_{i}|\geq K$ . Let $N_{i}^{\prime}\subseteq N_{i}$ with $|N_{i}^{\prime}|=K$ . Recall that $K=\sigma(7)$ . Now (8) follows from the definition of an $m$ -amicable class with $D_{1}=P$ , $D_{2}=B$ and $Y=N_{i}^{\prime}$ . This proves (8).

We may assume that for every $i$ there is a component $D_{i}$ of $G\setminus N[Z_{i}\cup Y_{i}]$ with $w(D_{i})>\frac{1}{2}$ , for otherwise the conclusion of Theorem 8.1 holds setting $Y=Z_{i}\cup Y_{i}$ . By (8), either $D_{i}$ is anticomplete to $L_{i}$ , or $D_{i}$ is anticomplete to $R_{i}$ . Let us say that $Q_{i}$ is of type $L$ if $D_{i}$ is anticomplete to $L_{i}$ , and that $Q_{i}$ is of type $R$ if $D_{i}$ is anticomplete to $R_{i}$ . By (8), every $Q_{i}$ is of at least one type.

(33) $Q_{1}$ is of type $L$ .

By (8), $D_{1}\cap(T\cup N[p_{k}])\neq\emptyset$ . It follows that $D_{1}$ contains a vertex $t$ such that $t\in N[p_{k}]$ or $t\in T\setminus N[Z_{1}]$ . Since $T\cap N(\{p_{1},\dots,p_{\operatorname{end}(1)}\})\subseteq N[Z_{1}]$ , and since $L_{1}\cup M_{1}$ is a subpath of $p_{1}\hbox{-}P\hbox{-}p_{\operatorname{end}(1)}$ , it follows that $D_{1}$ contains a vertex of $N[R_{1}]$ . Consequently, $Q_{1}$ is not of type $R$ , and so $Q_{1}$ is of type $L$ . This proves (8).

(34) We may assume that $Q_{r}$ is of type $R$ .

Suppose that $Q_{r}$ is of type $L$ . Then $D_{r}$ is anticomplete to $L_{r}$ . We claim that the set $Y=Z_{r}\cup Y_{r}\cup Y_{0}$ satisfies the conclusion of the theorem. Let $D$ be a component of $G\setminus N[Y]$ , and suppose that $w(D)>\frac{1}{2}$ . By (8) there exists $t\in D\cap(T\cup N[p_{k}])$ . Since $Z_{r}\cup Y_{r}\subseteq Y$ , we have that $D\subseteq D_{r}$ . It follows that $D$ is anticomplete to $L_{r}$ , and so $t\not\in N(L_{r})$ . Since $T\cap N(\{p_{r},\dots,p_{\operatorname{end}(r)}\})\subseteq N[Z_{r}]$ , it follows that $t\not\in\{p_{r},\dots,p_{\operatorname{end}(r)}\}$ . We deduce that $t\in N(\{p_{\operatorname{end}(r)+1},\dots,p_{k}\})$ . But then $t\in N[Y_{0}]$ , contrary to the fact that $Y_{0}\subseteq Y$ . This proves the claim, and (8) follows.

From now on we make the assumption of (8). By (8) we can choose $j$ minimum such that $Q_{j}$ is of type $R$ . By (8), $j>1$ . Let

Y=Z_{j-1}\cup Y_{j-1}\cup Z_{j}\cup Y_{j}\cup Y_{0}.

Then $|Y|\leq 6k+2m+1=d$ . We claim that $w(D)\leq\frac{1}{2}$ for every component $D$ of $G\setminus N[Y]$ . Suppose not, and let $D$ be a component of $G\setminus N[Y]$ with $w(D)>\frac{1}{2}$ . Since $Z_{j-1}\cup Y_{j-1}\subseteq Y$ , we have that $D\subseteq D_{j-1}$ . Since $Q_{j-1}$ is of type $L$ , it follows that $D$ is anticomplete to $L_{j-1}$ . Since $Z_{j}\cup Y_{j}\subseteq Y$ , we have that $D\subseteq D_{j}$ . Since $Q_{j}$ is of type $R$ , it follows that $D$ is anticomplete to $R_{j}$ . Since $T\cap N(\{p_{j-1},\dots,p_{\operatorname{end}(j)}\})\subseteq N[Z_{j-1}\cup Z_{j}]$ and since $p_{k}\in Y_{0}$ , we deduce that $D\cap(T\cup N[p_{k}])=\emptyset$ . But then $w(D)<\frac{1}{2}$ by (8), a contradiction. ∎

9. From balanced to small

The second step in the proof of Theorem 1.5 is to transform balanced separators with small domination number into balanced separators of small size. We show:

Theorem 9.1.

Let $t$ be an integer, let $d$ be as in Theorem 8.1 and let $c_{t}$ be as in Theorem 1.3. Let $G\in\mathcal{C}_{t}$ and let $w$ be a weight function on $G$ . Then there exists $Y\subseteq V(G)$ such that

•

$|Y|\leq 3c_{t}d\log n+t$ , and
•

$Y$ is a $w$ -balanced separator in $G$ .

The proof uses Theorems 1.3 and 8.1. In order to prove Theorem 9.1, we first prove a more general statement (below). We will then explain how to deduce Theorem 9.1 from it.

Theorem 9.2.

Let $L>0$ be an integer, let $G$ be a graph and let $w$ be a weight function on $G$ . Assume that there is no $L$ -banana in $G$ . There exists a clique $K$ in $G$ with the following property. Let $X\subseteq V(G)\setminus K$ be such that $w(D)\leq\frac{1}{2}$ for every component $D$ of $G\setminus(K\cup N[X])$ . Then there exists a balanced separator $Y$ in $G$ such that $|Y\setminus K|\leq 3L|X|$ .

Proof.

Let $(T,\chi)$ be a $3L$ -atomic tree decomposition of $G$ . By Theorems 2.2 and 2.3, we have that $(T,\chi)$ is tight and $3L$ -lean. By Theorem 2.7, there exists $t_{0}\in T$ such that $t_{0}$ is a center for $T$ . A vertex $v\in V(G)$ is $t_{0}$ -friendly if $v$ is not separated from $\chi(t_{0})\setminus v$ by a set of size $<3L$ . We show that $t_{0}$ -friendly vertices have the following important property.

(35) If $u,v\in\chi(t_{0})$ are not $t_{0}$ -friendly, then $u$ is adjacent to $v$ .

Suppose that $u$ and $v$ are not $t_{0}$ -friendly and that $u$ is non-adjacent to $v$ . Since there is no $L$ -banana in $G$ , Theorem 2.1 implies that there exists a set $X$ with $|X|<L$ such $X$ separates $u$ from $v$ . But this contradicts Theorem 2.4. This proves (9).

Let $K$ be the set of all the vertices of $G$ that are not $t_{0}$ -friendly. By (9), $K$ is a clique. We show that $K$ satisfies the conclusion of Theorem 9.2. Let $X\subseteq G\setminus K$ be such that $w(D)\leq\frac{1}{2}$ for every component $D$ of $G\setminus(K\cup N[X])$ .

For every $v\in G\setminus K$ , define the set $\Delta(v)$ as follows. Let $X_{v}\subseteq V(G)$ be such that $X_{v}$ separates $v$ from $\chi(t_{0})\setminus v$ , chosen with $|X_{v}|$ minimum. Let $\Delta(v)=X_{v}\cup\{v\}$ . Then $|\Delta(v)|\leq 3L$ and $N_{\chi(t_{0})}(v)\subseteq\Delta(v)$ . Let

Y=\bigcup_{v\in X}\Delta(v).

It follows that $|Y|\leq 3L|X|$ .

To complete the proof it is enough to show that $Y\cup K$ is a $w$ -balanced separator of $G$ . Suppose not, and let $D$ be a component of $G\setminus(Y\cup K)$ with $w(D)>\frac{1}{2}$ .

(36) If $D\cap(G_{t_{0}\rightarrow t}\setminus\chi(t_{0}))\neq\emptyset$ for some $t\in N_{T}(t_{0})$ , then $X$ is anticomplete to $D$ .

Suppose that $v\in X$ has a neighbor $d$ in $D$ . Let $(A,X_{v},B)$ be a separation such that $\chi(t_{0})\setminus X_{v}\subseteq B$ and $v\in A$ . Since $d\not\in X_{v}$ and $d$ is adjacent to $v$ , it follows that $d\in A$ . But then, since $D$ is a component of $G\setminus(K\cup Y)$ and $X_{v}\subseteq Y$ , it follows that $D\subseteq A$ . Consequently, $D\cap\chi(t_{0})=\emptyset$ . It follows that $D\subseteq G_{t_{0}\rightarrow t}\setminus\chi(t_{0})$ , and so $w(D)\leq w(G_{t_{0}\rightarrow t}\setminus\chi(t_{0}))\leq\frac{1}{2}$ , a contradiction. This proves (9).

(37) $D\cap N[X]=\emptyset$ .

Suppose there is $v\in X$ such that $v$ has a neighbor $d\in D$ . By (9), $d\in\chi(t_{0})$ . But then $d\in\Delta(v)\subseteq Y$ , a contradiction. This proves (9).

By (9), $D$ is a component of $G\setminus(K\cup N[X])$ , and therefore $w(D)\leq\frac{1}{2}$ , a contradiction. ∎

We now prove Theorem 9.1.

Proof.

Let $K$ be as in Theorem 9.2. Let $c_{t}$ be as in Theorem 1.3 and let $L=c_{t}\log n$ . Let $d$ be as Theorem 8.1. Define $w^{\prime}:V(G)\setminus K\rightarrow[0,1]$ as $w^{\prime}(v)=\frac{w(v)}{1-w(K)}$ . Then $w^{\prime}$ is a normal weight function on $G\setminus K$ . By Theorem 8.1, there exists $X\subseteq V(G)\setminus K$ such that

•

$|X|\leq d$ , and
•

$N[X]$ is a $w^{\prime}$ -balanced separator in $G\setminus K$ .

It follows that $w(D)\leq\frac{1}{2}$ for every component $D$ of $G\setminus(N[X]\cup K)$ . Now Theorem 9.2 applied with $L=c_{t}\log_{n}$ implies that there exists a $w$ -balanced separator $Y$ in $G$ such that $|Y\setminus K|\leq 3L|X|=3c_{t}(\log n)|X|\leq 3c_{t}d\log n$ . Since $G\in\mathcal{C}_{t}$ , it follows that $|Y|\leq 3c_{t}d\log n+t$ . ∎

10. The proof of Theorem 1.5

We are now ready to complete the proof of Theorem 1.5. The following result was originally proved by Robertson and Seymour in [33], and tightened by Harvey and Wood in [24]. It was then restated and proved in the language of $(w,c)$ -balanced separators in an earlier paper of this series [4].

Theorem 10.1 (Robertson, Seymour [33], see also [4, 24]).

Let $G$ be a graph, let $c\in[\frac{1}{2},1)$ , and let $d$ be a positive integer. If $G$ has a $(w,c)$ -balanced separator of size at most $d$ for every normal weight function $w:V(G)\to[0,1]$ , then $\operatorname{tw}(G)\leq\frac{1}{1-c}k$ .

We prove the following, which immediately implies Theorem 1.5.

Theorem 10.2.

Let $t$ be an integer. Let let $c_{t}$ be as in Theorem 1.3 and let $d$ be as in Theorem 1.4. Then every $G\in\mathcal{C}_{t}$ satisfies $\operatorname{tw}(G)\leq 6c_{t}d\log n+2t$ .

Proof.

By Theorem 9.1, for every normal weight function $w$ of $G$ there exists $w$ -balanced separator of size at most $3c_{t}d\log n+t$ . Now the result follows immediately from Lemma 10.1. ∎

11. Algorithmic consequences

We now summarize the algorithmic consequences of our structural results, specifically of Theorems 1.4 and 1.5.

The consequences for graphs in $\mathcal{C}_{t}$ are the most immediate. In particular, using the factor $2$ approximation algorithm of Korhonen [26] (or the simpler factor $4$ approximation algorithm of Robertson and Seymour [34]) for treewidth, we can compute a tree decomposition of width at most $O(c_{t}\log n)$ in time $2^{O(c_{t}\log n)}n^{O(1)}=n^{O(c_{t})}$ . A number of well-studied graph problems, including Stable Set, Vertex Cover, Feedback Vertex Set Dominating Set and $r$ -Coloring (for fixed $r$ ) admit algorithms with running time $2^{O(k)}n$ when a tree decomposition of $G$ of width $k$ is provided as input (See, for example, [16] chapters 7 and 11, as well as [10]). Since $2^{O(c_{t}\log n)}=n^{O(c_{t})}$ this immediately leads to polynomial time algorithms for these problems in $\mathcal{C}_{t}$ .

Theorem 11.1.

Let $t\geq 1$ be fixed and P be a problem which admits an algorithm running in time $\mathcal{O}(2^{\mathcal{O}(k)}|V(G)|)$ on graphs $G$ with a given tree decomposition of width at most $k$ . Then P is solvable in time $n^{\mathcal{O}(t)}$ in $\mathcal{C}_{t}$ . In particular, Stable Set, Vertex Cover, Feedback Vertex Set, Dominating Set and $r$ -Coloring (with fixed $r$ ) are all polynomial-time solvable in $\mathcal{C}_{t}$ .

This list of problems is far from exhaustive, it is worth mentioning the work of Pilipczuk [32] who provides a relatively easy-to-check sufficient condition (expressibility in Existential Counting Modal Logic) for a problem to admit such an algorithm.

Theorem 11.1 implies a polynomial time algorithm for $r$ -Coloring (with fixed $r$ ) in $\mathcal{C}$ (without any assumptions on max clique size) because every graph that contains a clique of size $r+1$ can not be $r$ -colored. Thus, to solve $r$ -Coloring we can first check for the existence of an $(r+1)$ -clique in time $n^{r+\mathcal{O}(1)}$ . If an $(r+1)$ -clique is present, report that no $r$ -coloring exists, otherwise invoke the algorithm of Theorem 11.1 with $t=r+1$ . This yields the following result.

Theorem 11.2.

For every positive integer $r$ , $r$ -Coloring is polynomial-time solvable in $\mathcal{C}$ .

Let us now discuss another important problem, and that is Coloring. It is well-known (and also follows immediately from Theorem 7.1), that for every $t$ there exists a number $d(t)$ such that all graphs in $\mathcal{C}_{t}$ have chromatic number at most $d(t)$ . Also, for each fixed $r$ , by Theorem 11.1, $r$ -Coloring is polynomial-time solvable in $\mathcal{C}_{t}$ . Now by solving $r$ -Coloring for every $r\in\{1,\dots,d(t)\}$ , we also obtain a polynomial-time algorithm for Coloring in $\mathcal{C}_{t}$ .

Theorem 11.1 quite directly leads to a polynomial-time approximation scheme (PTAS) for several problems on graphs in $\mathcal{C}$ . We illustrate this using Vertex Cover as an example.

Theorem 11.3.

There exists an algorithm that takes as input a graph $G\in\mathcal{C}$ and $0<\epsilon\leq 1$ , runs in time $n^{O(c_{2/\epsilon})}$ and outputs a vertex cover of size at most $(1+\epsilon)\textsf{vc}(G)$ , where $\textsf{vc}(G)$ is the size of the minimum vertex cover of $G$ .

Proof.

Check in time $n^{O(1/\epsilon)}$ whether $G$ contains as input a clique $C$ of size at least $2/\epsilon$ . If $G$ does not have a clique of size at least $2/\epsilon$ then compute an optimal vertex cover of $G$ in time $n^{O(c_{2/\epsilon})}$ using the algorithm of Theorem 11.1. If $G$ contains such a clique $C$ then run the algorithm recursively on $G-C$ to obtain a vertex cover $S$ of $G-C$ of size at most $(1+\epsilon)\textsf{vc}(G-C)$ . Return $S\cup C$ ; clearly $S\cup C$ is a vertex cover of $G$ . Furthermore, every vertex cover of $G$ (and in particular a minimum one) contains at least $|C|-1$ vertices of $C$ . It follows that $\textsf{vc}(G-C)\leq\textsf{vc}(G)-|C|+1$ and that therefore

	$\displaystyle\|S\cup C\|$	$\displaystyle=\|S\|+\|C\|$
		$\displaystyle\leq(1+\epsilon)\textsf{vc}(G-C)+\|C\|$
		$\displaystyle\leq(1+\epsilon)(\textsf{vc}(G)-\|C\|+1)+\|C\|$
		$\displaystyle\leq(1+\epsilon)(\textsf{vc}(G))$

Here the last inequality follows because for $C\geq 2/\epsilon$ it holds that $(1+\epsilon)(|C|-1)\geq|C|$ . ∎

The PTAS of Theorem 11.3 generalizes without modification (except for the constant $2$ in the $2/\epsilon$ ) to Feedback Vertex Set, and more generally, to deletion to any graph class which is closed under vertex deletion, excludes some clique, and admits an algorithm (for the deletion problem) with running time $2^{O(k)}n^{O(1)}$ on graphs of treewidth $k$ . Despite the tight connection between Vertex Cover and Stable Set, Theorem 11.3 does not directly lead to a PTAS for Stable Set on graphs in $\mathcal{C}$ . On the other hand, a QPTAS for Stable Set in $\mathcal{C}$ follows from Theorem 1.4 together with an argument of Chudnovsky, Pilipczuk, Pilipczuk and Thomassé [14], who gave a QPTAS for Stable Set in $P_{k}$ -free graphs (they refer to the problem as Independent Set). For ease of reference, we repeat their argument here.

Theorem 11.4.

There exists an algorithm that takes as input a graph $G\in\mathcal{C}$ and $0<\epsilon\leq 1$ , runs in time $n^{O(\log^{2}n/\epsilon)}$ and outputs a stable set $S$ in $G$ of size at least $(1-\epsilon)\alpha(G)$ , where $\alpha(G)$ is the size of the maximum size stable set in $G$ .

Proof.

The algorithm is recursive, taking as input the graph $G$ and also an integer $N$ . Initially, the algorithm is called with $N=|V(G)|$ , in all subsequent recursive calls the value of $N$ remains the same. If $G$ is a single vertex the algorithm returns $V(G)$ . If $G$ is disconnected, the algorithm runs itself recursively on each of the connected components and returns the union of the stable sets returned for each of them.

Let $d$ be the constant of Theorem 1.4. If $G$ is a connected graph on at least two vertices the algorithm iterates over all subsets $S\subseteq V(G)$ of size at most $\frac{2d\log n\log N}{\epsilon}$ and all subsets $Y$ of size at most $d$ such that each connected component of $G-(N(S)\cup N[Y])$ has at most $|V(G)|/2$ vertices. For each such pair $(S,Y)$ the algorithm calls itself recursively on $G-(N(S)\cup N[Y])$ and returns the maximum size stable set found over all such recursive calls.

Clearly, the set returned by the algorithm is a stable set. For each pair $(S,Y)$ which results in a recursive call of the algorithm, each connected component of the graph $G-(N(S)\cup N[Y])$ that the algorithm is called on has at most $|V(G)|/2$ vertices. Therefore the running time of the algorithm is governed by the recurrence $T(n)\leq n^{\frac{2d\log n\log N}{\epsilon}+d}\cdot n\cdot T(n/2)$ which solves to $T(n)\leq n^{O(\frac{d\log^{2}n\log N}{\epsilon})}$ . Since the algorithm is initially called with $N=|V(G)|$ the running time of the algorithm is upper bounded by $|V(G)|^{O(\frac{d\log^{3}|V(G)|}{\epsilon})}$ .

It remains to argue that the size of the stable set output by the algorithm is at least $(1-\epsilon)\alpha(G)$ . To that end, we show that the size of the independent set output by a recursive call on $(G,N)$ is at least $\alpha(G)(1-\epsilon\frac{\log|V(G)|}{\log N})$ . Let $I$ be a maximum size stable set of $G$ . By a standard coupon-collector argument (see e.g. Lemma 4.1 in [14]) there exists a choice of $S\subseteq I$ of size at most $\frac{d\log N}{\epsilon}2\log n$ such that no vertex of $G-N(S)$ has at least $|I|\frac{\epsilon}{d\log N}$ neighbors in $I$ . Let $Y$ now be the set (of size at most $d$ ) given by Lemma 1.4 applied to $G-N[S]$ . Since no vertex in $Y$ has at least $\frac{\epsilon}{d\log N}$ neighbors in $I$ it follows that $|I-(N(S)\cup N[Y])|\geq|I|(1-\frac{\epsilon}{\log N})$ . Furthermore, each connected component of $G-(N(S)\cup N[Y])$ has at most $|V(G)|/2$ vertices. By the inductive hypothesis the recursive call on $(G-(N(S)\cup N[Y]),N)$ outputs a stable set of size at least

|I|\left(1-\frac{\epsilon}{\log N}\right)\left(1-\frac{\epsilon(\log|V(G)|-1)}{\log N}\right)\geq|I|\left(1-\epsilon\frac{\log|V(G)|}{\log N}\right)\mbox{.}

Since $\alpha(G)=|I|$ the recursive call on $(G,N)$ outputs a stable set of size at least $\alpha(G)(1-\epsilon\frac{\log|V(G)|}{\log N})$ , as claimed. ∎

12. Acknowledgments

We are grateful to Tara Abrishami and Bogdan Alecu for their involvement in early stages of this work. We also thank Kristina Vušković for many years of conversations about the structure of even-hole-free graphs.

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	$\displaystyle\|S\cup C\|$	$\displaystyle=\|S\|+\|C\|$
		$\displaystyle\leq(1+\epsilon)\textsf{vc}(G-C)+\|C\|$
		$\displaystyle\leq(1+\epsilon)(\textsf{vc}(G)-\|C\|+1)+\|C\|$
		$\displaystyle\leq(1+\epsilon)(\textsf{vc}(G))$

Induced subgraphs and tree decompositions XV. Even-hole-free graphs with bounded clique number have logarithmic treewidth

Abstract.

1. Introduction

Conjecture 1.1 (Sintiari and Trotignon [36]).

Conjecture 1.2 (Sintiari and Trotignon [36]).

Theorem 1.3.

Theorem 1.4.

Theorem 1.5.

1.1. Proof outline and organization

2. Special tree decompositions and connectivity

2.1. Connectivity

Theorem 2.1 (Menger [30]).

2.2. Lean tree decompositions

Theorem 2.2 (Bellenbaum and Diestel [9], see Carmesin, Diestel, Hamann, Hundertmark [13], see also Weißauer [38]).

Theorem 2.3 (Weißauer [38]).

Theorem 2.4.

Proof.

Theorem 2.5 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

2.3. Catching a banana

Theorem 2.6.

Proof.

2.4. Centers of tree decompositions.

Theorem 2.7.

3. Wheels and star cutsets

Theorem 3.1 (Abrishami, Chudnovsky, Vušković [6]).

Theorem 3.2 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

Theorem 3.3.

Theorem 3.4 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

4. Stable sets of safe hubs

Lemma 4.1.

Proof.

Lemma 4.2 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

Lemma 4.3.

Proof.

Theorem 4.4 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

Lemma 4.5.

Proof.

Lemma 4.6.

Proof.

Lemma 4.7.

Proof.

Lemma 4.8.

Lemma 4.9.

Theorem 4.10.

Proof.

Theorem 4.11.

Proof.

Theorem 4.12.

Proof.

5. Connectifiers

Theorem 5.1 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

Theorem 5.2.

Proof.

Theorem 5.3.

Theorem 5.4 (Erdős and Szekeres [20]).

Lemma 5.5.

Proof.

Lemma 5.6.

Proof.

Proof.

Theorem 5.7.

Proof.

Theorem 5.8 (Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl [3]).

6. Bounding the number of non-hubs in a minimal separator

Theorem 6.1 (Lozin and Razgon [29]).

Lemma 6.2.

Theorem 6.3.

Proof.

7. The proof of Theorem 1.3

Theorem 7.1 (Abrishami, Chudnovsky, Hajebi, Spirkl [5]).

Theorem 7.2.

Proof.

8. Dominated balanced separators

Theorem 8.1.

Theorem 8.2 (Addario-Berry, Chudnovsky, Havet, Reed, Seymour [7], da Silva, Vušković [17]).

Theorem 8.3.

Proof.

Theorem 8.4.

Proof.

Lemma 8.5 (Chudnovsky, Pilipczuk, Pilipczuk, Thomassé [14], Lemma 5.3).

Induced subgraphs and tree decompositions
XV. Even-hole-free graphs with bounded clique number have logarithmic treewidth