Helly-gap of a graph and vertex eccentricities

All graphs in this paper are unweighted, undirected, connected, and without loops or multiple edges. In a graph $G=(V(G),E(G))$, a disk $D_{G}(v,r)$ with radius $r$ and centered at a vertex $v$ consists of all vertices with distance at most $r$ from $v$, i.e., $D_{G}(v,r)=\{u\in V(G):d_{G}(u,v)\leq r\}$. A graph $G$ is called Helly if every system of pairwise intersecting disks of $G$ has a non-empty intersection.

Helly graphs are well investigated. They have several characterizations and important features as established in [16, 17, 4, 44, 46, 5]. They are exactly the so-called absolute retracts of reflexive graphs and possess a certain elimination scheme [16, 17, 4, 44, 5]. They can be recognized in $O(n^{2}m)$ time [16] where $n$ is the number of vertices and $m$ is the number of edges. The Helly property works as a compactness criterion on graphs [46]. More importantly, every graph is isometrically embeddable into a Helly graph [36, 27, 40] (see Section 3 for more details).

Many nice properties of Helly graphs are based on the eccentricity $e_{G}(v)$ of a vertex $v$, which is defined as the maximum distance from $v$ to any other vertex of the graph (i.e., $e_{G}(v)=\max_{u\in V(G)}d_{G}(v,u)$). The minimum (maximum) eccentricity in a graph $G$ is the radius $rad(G)$ (diameter $diam(G)$). The center $C(G)$ of a graph $G$ is the set of all vertices with minimum eccentricity (i.e., $C(G)=\{v\in V(G):e_{G}(v)=rad(G)\}$. Graph’s diameter is tightly bounded in Helly graphs as $2rad(G)\geq diam(G)\geq 2rad(G)-1$ [16, 17]. Moreover, the eccentricity function in Helly graphs is unimodal [17], that is, any local minimum coincides with the global minimum; this is equivalent to the condition that, for any vertex $v\in V(G)$, $e_{G}(v)=d_{G}(v,C(G))+rad(G)$ holds [16]. The unimodality of the eccentricity function was recently used in [29, 28] to compute the radius, diameter and a central vertex of a Helly graph in subquadratic time. For a vertex $v\in V(G)$, let $F_{G}(v)$ denote the set of all vertices farthest from $v$, that is, $F_{G}(v)=\{u\in V(G):e_{G}(v)=d_{G}(v,u)\}$. For every vertex $v$ of a Helly graph $G$, each vertex $u\in F_{G}(v)$ satisfies $e_{G}(u)\geq 2rad(G)-diam(C(G))$ [16]. Although the center $C(G)$ of a Helly graph $G$ may have an arbitrarily large diameter (as any Helly graph is the center of some Helly graph), $C(G)$ induces a Helly graph and is isometric in $G$ [16]. Additionally, any power of a Helly graph is a Helly graph as well [16].

In this paper, we introduce a far reaching generalization of Helly graphs, $\alpha$-weakly-Helly graphs. We define $\alpha$-weakly-Helly graphs as those graphs which are “weakly” Helly with the following simple generalization: for any system of disks, if the disks pairwise intersect, then by expanding the radius of each disk by some integer $\alpha$ there forms a common intersection. Thus, Helly graphs are exactly 0-weakly-Helly graphs.

Interestingly, there are a few results in the literature which demonstrate that such a $\alpha$-weakly-Helly property with bounded $\alpha$ is present in some graphs and in some metric spaces. In [14], Chepoi and Estellon showed that for every $\delta$-hyperbolic geodesic space $(X,d)$ (and for every $\delta$-hyperbolic graph $G$), if disks of the family $\mathcal{F}=\{D(x,r(x)):v\in S\subseteq X,S$ is compact$\}$ pairwise intersect, then the disks $\{D(x,r(x)+2\delta):x\in S\}$ have a nonempty common intersection (see also  [11]). That is, the disks in $\delta$-hyperbolic geodesic spaces and in $\delta$-hyperbolic graphs satisfy $(2\delta)$-weakly-Helly property. Even earlier, Lenhart, Pollack et al. [41, Lemma 9] established that the disks of simple polygons endowed with the link distance satisfy a Helly-type property which implies 1-weakly-Helly property. For chordal graphs [20] and for distance-hereditary graphs [18] (as well as for a more general class of graphs [10]) the following similar result is known: if for a set $M\subseteq V(G)$ and radius function $r:M\rightarrow\mathbb{N}$, every two vertices $x,y\in M$ satisfy $d_{G}(x,y)\leq r(x)+r(y)+1$, then there is a clique $K$ in $G$ such that $d_{G}(v,K)\leq r(v)$ for all $v\in M$. Hence, clearly, every chordal graph and every distance-hereditary graph is 1-weakly-Helly. Note that two disks $D_{G}(v,p)$ and $D_{G}(u,q)$ intersect if and only if $d_{G}(v,u)\leq p+q$.

There are numerous other approaches to generalize in some form the Helly graphs. One may loosen the restriction on the type of sets which must satisfy the Helly property. If one considers neighborhoods or cliques, rather than arbitrary disks, then one gets the neighborhood-Helly graphs and the clique-Helly graphs as superclasses of Helly graphs (see [42, 34, 6] and papers cited therein). One may also generalize the Helly-property that a family of sets satisfies. This can be accomplished by specifying a minimum number or size of sets that pairwise intersect, a minimum number of subfamilies that have a common intersection, or the size of the intersection (see [30] and papers cited therein). Far reaching examples include the $(p,q)$-Helly property (see [3, 2] and papers cited therein) and the fractional Helly property (see [37, 7] and papers cited therein). There is also a related to ours notion of $\lambda$-hyperconvexity in metric spaces $(X,d)$ [31, 38], where $\lambda$ is the smallest multiplicative constant such that for any system of disks $\mathcal{F}=\{D(x,r(x)):x\in S\subseteq X\}$, the following property holds: if $X\cap Y\neq\emptyset$ for every $X,Y\in\mathcal{F}$, then $\underset{x\in S}{\bigcap}D(x,\lambda\cdot r(x))\neq\emptyset$. Several classical metric spaces are $\lambda$-hyperconvex for a bounded $\lambda$ (see [31] and papers cited therein). These include reflexive Banach spaces and dual Banach spaces ($\lambda\leq 2$) and Hilbert spaces ($\lambda\leq\sqrt{2}$). The classical Jung Theorem asserting that each subset $S$ of the Euclidean space $\mathbb{E}^{m}$ with finite diameter $D$ is contained in a disk of radius at most $\sqrt{\frac{m}{2(m+1)}}D$ belongs to this kind of results, too.

In this paper, we are interested in $\alpha$-weakly-Helly graphs, where $\alpha$ describes an additive constant by which radius of each disk in the family of pairwise intersecting disks can be increased in order to obtain a nonempty common intersection of all expanded disks. By definition, any $\alpha$-weakly-Helly graph is $(\alpha+1)$-hyperconvex. We find that there are also an abundance of graph classes with a bounded Helly-gap $\alpha(G)$. Here, we generalize many eccentricity related results known for Helly graphs (as well as for chordal graphs, distance-hereditary graphs and $\delta$-hyperbolic graphs) to all graphs, parameterized by their Helly-gap $\alpha(G)$. We provide a characterization of weakly-Helly graphs with respect to their injective hulls and use this as a tool to prove those generalizations. Several additional well-known graph classes that are $\alpha$-weakly-Helly for some constant $\alpha$ are also identified.

The main results of this paper can be summarized as follows. After providing in Section 2 necessary definitions, notations and some relevant results on Helly graphs, in Section 3 we give a characterization of $\alpha$-weakly-Helly graphs through distances in injective hulls. The injective hull of a graph $G$, denoted by $\mathcal{H}(G)$, is a (unique) minimal Helly graph which contains $G$ as an isometric subgraph [36, 27]. We show that $G$ is an $\alpha$-weakly-Helly graph if and only if for every vertex $h\in V(\mathcal{H}(G))$ there is a vertex $v\in V(G)$ such that $d_{\mathcal{H}(G)}(h,v)\leq\alpha$ (Theorem 1). In Section 4, we relate the diameter, radius, and all eccentricities in $G$ to their counterparts in $H:=\mathcal{H}(G)$. In particular, we show that $e_{G}(v)=e_{H}(v)$ for all $v\in V(G)$, $diam(G)=diam(H)$, and $rad(G)-\alpha(G)\leq rad(H)\leq rad(G)$. Additionally, we show $diam_{G}(C(G))-2\alpha(G)\leq diam_{H}(C(H))\leq diam_{G}(C^{\alpha(G)}(G))+2\alpha(G)$, where $C^{\ell}(G)=\{v\in V(G):e_{G}(v)\leq rad(G)+\ell\}$. We also provide several bounds on $\alpha(G)$ including its relation to diameter and radius as well as investigate the Helly-gap of powers of weakly-Helly graphs (Theorem 3). In particular, $\lfloor(2rad(G)-diam(G))/2\rfloor\leq\alpha(G)\leq\lfloor diam(G)/2\rfloor$ holds. The eccentricity function in $\alpha$-weakly-Helly graphs is shown to exhibit the property that any vertex $v\notin C^{\alpha}(G)$ has a nearby vertex, within distance $2\alpha+1$ from $v$, with strictly smaller eccentricity (Theorem 4). In Section 5, we give upper and lower bounds on the eccentricity $e_{G}(v)$ of a vertex $v$. We consider bounds based on the distance from $v$ to a closest vertex in $C^{\alpha(G)}(G)$, whether $v$ is farthest from some other vertex and if $diam_{G}(C^{\alpha(G)}(G))$ is bounded. In particular, we show that $|e_{G}(v)-d_{G}(v,C^{\alpha(G)}(G))-rad(G)|\leq\alpha(G)$ holds for any vertex $v\in V(G)$ (Theorem 5). We also prove the existence of a spanning tree $T$ of $G$ which gives an approximation of all vertex eccentricities in $G$ with an additive error depending only on $\alpha(G)$ and $diam_{G}(C^{\alpha(G)}(G))$ (Theorem 7). We find that in any shortest path to $C^{\alpha(G)}(G)$, the number of vertices with locality more than 1 does not exceed $2\alpha(G)$, where the locality of a vertex $v$ is the minimum distance from $v$ to a vertex of strictly smaller eccentricity (Theorem 8). All these results greatly generalize some known facts about distance-hereditary graphs, chordal graphs, and $\delta$-hyperbolic graphs. Those graphs have bounded Helly gap. In Section 6, we identify several more (well-known) graph classes with a bounded Helly-gap, including $k$-chordal graphs, AT-free graphs, rectilinear grids, graphs with a bounded $\alpha_{i}$-metric, and graphs with bounded tree-length or tree-breadth. We conclude with open questions in Section 7.

All graphs $G=(V(G),E(G))$ occurring in this paper are undirected, connected, and without loops or multiple edges. The length of a path from a vertex $u$ to a vertex $v$ is the number of edges in the path. The distance $d_{G}(u,v)$ between two vertices $u$ and $v$ is the length of a shortest path connecting them in $G$. The distance from a vertex $u$ to a vertex set $S\subseteq V(G)$ is defined as $d_{G}(u,S)=\min_{s\in S}d_{G}(u,s)$. The eccentricity of a vertex is defined as $e_{G}(v)=\max_{u\in V(G)}d_{G}(v,u)$. The vertex set $F_{G}(v)$ for a vertex $v\in V(G)$ denotes the set of all vertices farthest from $v$, that is, $F_{G}(v)=\{u\in V(G):e_{G}(v)=d_{G}(v,u)\}$. A disk of radius $k$ centered at a set $S$ (or a vertex) is the set of vertices of distance at most $k$ from $S$, that is, $D_{G}(S,k)=\{u\in V(G):d_{G}(u,S)\leq k\}$. We omit $G$ from subindices when the graph is known by context.

The $k^{th}$ power $G^{k}$ of a graph $G$ is a graph that has the same set of vertices, but in which two distinct vertices are adjacent if and only if their distance in $G$ is at most $k$. A subgraph $G^{\prime}$ of a graph $G$ is called isometric (or a distance preserving subgraph) if for any two vertices $x,y$ of $G^{\prime}$, $d_{G}(x,y)=d_{G^{\prime}}(x,y)$ holds. The interval $I(x,y)$ is the set of all vertices belonging to a shortest $(x,y)$-path, that is, $I(x,y)=\{u\in V(G):d(x,u)+d(u,y)=d(x,y)\}$. The interval slice $S_{k}(x,y)$ is the set of vertices on $I(x,y)$ which are at distance $k$ from vertex $x$. An interval $I(x,y)$ is said to be $\kappa$-thin if, for any $0\leq\ell\leq d(x,y)$ and any $u,v\in S_{\ell}(x,y)$, $d(u,v)\leq\kappa$ holds. The smallest $\kappa$ for which all intervals of $G$ are $\kappa$-thin is called the interval thinness of $G$ and denoted by $\kappa(G)$. By $G-\{x\}$ we denote an induced subgraph of $G$ obtained from $G$ by removing a vertex $x\in V(G)$.

The minimum (maximum) eccentricity in a graph $G$ is the radius $rad(G)$ (diameter $diam(G)$). The center $C(G)$ is the set of vertices with minimum eccentricity (i.e., $C(G)=\{v\in V(G):e(v)=rad(G)\}$. It will be useful to define also $C^{k}(G)=\{v\in V(G):e(v)\leq rad(G)+k\}$; then $C(G)=C^{0}(G)$. Let $M\subseteq V(G)$ be a subset of vertices of $G$. We distinguish the eccentricity with respect to $M$ as follows: denote by $e_{G}^{M}(v)$ the maximum distance from vertex $v$ to any vertex $u\in M$, i.e., $e_{G}^{M}(v)=\max_{u\in M}d_{G}(v,u)$. We then define $F_{G}^{M}(v)=\{u\in M:e_{G}^{M}(v)=d_{G}(v,u)\}$, $rad_{G}(M)=\min_{v\in V(G)}e_{G}^{M}(v)$, and $diam_{G}(M)=\max_{v\in M}e_{G}^{M}(v)$. Let $C_{G}^{\ell}(M)=\{v\in V(G):e_{G}^{M}(v)\leq rad_{G}(M)+\ell\}$ and $C_{G}(M)=C_{G}^{0}(M)$. For simplicity, when $M=V(G)$, we continue to use the notation $rad(G)$, $C(G)$, and $diam(G)$.

The following results for Helly graphs will prove useful for $\alpha$-weakly-Helly graphs.

Given this lemma, it will be convenient to also denote by $C_{G}(M)$ a subgraph of $G$ induced by $C_{G}(M)$.

For a graph $G$ and a subset $M\subseteq V(G)$, the eccentricity function $e_{G}^{M}(\cdot)$ is called unimodal if every vertex $v\in V(G)\setminus C_{G}(M)$ has a neighbor $u$ such that $e_{G}^{M}(u)<e_{G}^{M}(v)$.

The following two results were earlier proven in [16] only for $M=V(G)$ and then later extended in [21] to all $M\subseteq V(G)$.

Recall that the injective hull of $G$, denoted by $\mathcal{H}(G)$, is a minimal Helly graph which contains $G$ as an isometric subgraph [36, 27]. It turns out that $\mathcal{H}(G)$ is unique for every $G$ [36]. When $G$ is known by context, we often let $H:=\mathcal{H}(G)$. By an equivalent definition of an injective hull [27] (also called a tight span), each vertex $f\in V(\mathcal{H}(G))$ can be represented as a vector with nonnegative integer values $f(x)$ for each $x\in V(G)$, such that the following two properties hold:

Additionally, there is an edge between two vertices $f,g\in V(\mathcal{H}(G))$ if and only if their Chebyshev distance is 1, i.e., $\max_{x\in V(G)}\lvert f(x)-g(x)\rvert=1$. Thus, $d_{H}(f,g)=max_{x\in V(G)}\lvert f(x)-g(x)\rvert$. Notice that if $f\in V(\mathcal{H}(G))$, then $\{D(x,f(x)):x\in V(G)\}$ is a family of pairwise intersecting disks. For a vertex $z\in V(G)$, define the distance function $d_{z}$ by setting $d_{z}(x)=d_{G}(z,x)$ for any $x\in V(G)$. By the triangle inequality, each $d_{z}$ belongs to $V(\mathcal{H}(G))$. An isometric embedding of $G$ into $\mathcal{H}(G)$ is obtained by mapping each vertex $z$ of $G$ to its distance vector $d_{z}$.

We classify every vertex $v$ in $V(\mathcal{H}(G))$ as either a real vertex or a Helly vertex. A vertex $f\in V(\mathcal{H}(G))$ is a real vertex provided $f=d_{z}$ for some $z\in V(G)$, i.e., there is a one-to-one correspondence between $z\in V(G)$ and its representative real vertex $f\in V(\mathcal{H}(G))$ which uniquely satisfies $f(z)=0$ and $f(x)=d_{G}(z,x)$ for all $x\in V(G)$. When working with $\mathcal{H}(G)$, we will use interchangeably the notation $V(G)$ to represent the vertex set in $G$ as well as the vertex subset of $\mathcal{H}(G)$ which uniquely corresponds to the vertex set of $G$. Then, a vertex $v\in V(\mathcal{H}(G))$ is a real vertex if it belongs to $V(G)$ and a Helly vertex otherwise. Equivalently, a vertex $h\in V(\mathcal{H}(G))$ is a Helly vertex provided that $h(x)\geq 1$ for all $x\in V(G)$, that is, a Helly vertex exists only in the injective hull $\mathcal{H}(G)$ and not in $G$. We will show in Theorem 1 that a graph $G$ is $\alpha$-weakly-Helly if and only if the distance from any Helly vertex in $\mathcal{H}(G)$ to a closest real vertex in $V(G)$ is no more than $\alpha$. We recently learned that this result was independently discovered by Chalopin et al. [8]. They use name coarse Helly property for $\alpha$-weakly-Helly property used here.

The following properties will be useful to the main result of this section and to later sections. A vertex $x$ is called a peripheral vertex if $I(y,x)\not\subset I(y,z)$ for some vertex $y$ and all vertices $z\neq x$. We show next that the peripheral vertices of $\mathcal{H}(G)$ are real vertices. This adheres to the intuitive notion that an injective hull contains all of the Helly vertices “between” the vertices of $G$, so that the outermost vertices of $\mathcal{H}(G)$ are real.

Moreover, we show that any shortest path of $\mathcal{H}(G)$ is a subpath of a shortest path between real vertices, which will later prove a useful property of injective hulls.

We are now ready to prove the main result of this section.

The injective hull is also useful to prove that $\alpha$-weakly-Helly graphs are closed under pendant vertex addition.