Using edge cuts
to find Euler tours and Euler families in hypergraphs

It is common knowledge that a connected graph admits an Euler tour — that is, a closed walk traversing each edge of the graph exactly once — if and only if the graph has no vertices of odd degree. The notion of an Euler tour can be generalized to hypergraphs in the obvious way, and has been studied as such in [5, 1]. In addition, Bahmanian and Šajna [1] also introduced the notion of an Euler family, which is a family of closed walks that jointly traverse each edge of the hypergraph exactly once and cannot be concatenated. For a connected graph, an Euler family precisely corresponds to an Euler tour; for general connected hypergraphs, however, the two notions give rise to two rather distinct problems: Euler family, which is of polynomial complexity [1], and Euler tour, which is NP-complete [5]. The question of how to reduce the search for an Euler family or tour in a hypergraph to smaller hypergraphs is thus particularly pertinent in the case of Euler tours.

An Euler tour (family) is called spanning if it traverses every vertex of the hypergraph. In [6], Steimle and Šajna showed how to use certain vertex cuts to reduce the problem of finding a spanning Euler tour (family) in a hypergraph $H$ to some smaller hypergraphs derived from subhypergraphs of $H$. In the present paper, with the same goal in mind, we shall use edge cuts instead. This new approach represents a major improvement over the results from [6]. First, it can be used for general Euler tours and families, not just for spanning Euler tours and families. Second, while the main results from [6] apply only to hypergraphs with vertex cuts of cardinality at most two, the present approach applies to edge cuts of any size, and hence to all hypergraphs, as every non-trivial hypergraph has an edge cut. In addition, we introduce new elegant techniques of edge cut assignments and collapsed hypergraphs, which will be useful in problems beyond the scope of this paper.

After reviewing the terminology and providing some basic tools in Section 2, we shall introduce edge cuts in Section 3, and the first main tool — edge cut assignments — in Section 4. The key result of this section is Theorem 4.4, where we show that a hypergraph admits an Euler family (tour) if and only if there exists an edge cut assignment such that the associated hypergraph admits an Euler family (tour). This theorem forms the basis of all reduction results and algorithms.

In Section 5, we present our first main reduction result. Namely, in Theorems 5.2 and 5.3 we show that a hypergraph $H$ with a minimal edge cut $F$ admits an Euler family (tour) if and only if certain subhypergraphs of H (obtained from the connected components of $H{\backslash}F$) admit an Euler family (tour).

We begin Section 6 by introducing a collapsed hypergraph of a hypegraph $H$; that is, a hypergraph obtained from $H$ by “collapsing” (identifying) the vertices in a given subset of the vertex set of $H$. We then present our second main reduction result. In Theorem 6.2, we show that a hypergraph $H$ admits an Euler family if and only if certain collapsed hypergraphs obtained via an edge cut assignment admit Euler families. The analogous result for Euler tours is presented in Corollaries 6.3 and 6.4; the former contains the proof of necessity, while the latter contains the proof of sufficiency, but requires an additional assumption.

Each of Sections 5 and 6 concludes with a description of pertinent algorithms that determine whether or not a hypergraph admits an Euler family (tour). Since all of our proofs are constructive, the algorithms can easily be modified to construct an Euler family (tour) if one exists.

We begin with some basic concepts related to hypergraphs, which will be used in later discussions. For any graph- and hypergraph-theoretic terms not defined here, we refer the reader to [3] and [2], respectively.

A hypergraph $H$ is an ordered pair $(V,E)$, where $V$ is a non-empty finite set and $E$ is a finite multiset of $2^{V}$. To denote multisets, we shall use double braces, $\left\{\!\!\left\{.\right\}\!\!\right\}$. The elements of $V=V(H)$ and $E=E(H)$ are called vertices and edges, respectively. A hypergraph is said to be trivial if it has only one vertex, and empty if it has no edges. The size of a hypergraph $H=(V,E)$ is $|H|=\sum_{e\in E}|e|$.

Let $H=(V,E)$ be a hypergraph, and $u,v\in V$. If $u\neq v$ and there exists an edge $e\in E$ such that $u,v\in e$, then we say that $u$ and $v$ are adjacent (via the edge $e$); this is denoted $u\sim_{H}v$. If $v\in V$ and $e\in E$ are such that $v\in e$, then $v$ is said to be incident with $e$, and the ordered pair $(v,e)$ is called a flag of $H$. The set of flags of $H$ is denoted by $F(H)$. The degree of a vertex $v\in V$ is the number of edges in $E$ incident with $v$, and is denoted by $\deg_{H}(v)$, or simply $\deg(v)$ when there is no ambiguity.

A hypergraph $H^{\prime}=(V^{\prime},E^{\prime})$ is called a subhypergraph of the hypergraph $H=(V,E)$ if $V^{\prime}\subseteq V$ and $E^{\prime}=\left\{\!\!\left\{e\cap V^{\prime}:e\in E^{\prime\prime}\right\}\!\!\right\}$ for some submultiset $E^{\prime\prime}$ of $E$. For any subset $V^{\prime}\subseteq V$, we define the subhypergraph of $H$ induced by $V^{\prime}$ to be the hypergraph $(V^{\prime},E^{\prime})$ with $E^{\prime}=\left\{\!\!\left\{e\cap V^{\prime}:e\in E,e\cap V^{\prime}\neq\emptyset\right\}\!\!\right\}$. Thus, we obtain the subhypergraph induced by $V^{\prime}$ by deleting all vertices in $V-V^{\prime}$ from $V$ and from each edge of $H$, and subsequently deleting all empty edges. For any subset $E^{\prime}\subseteq E$, we denote the subhypergraph $(V,E-E^{\prime})$ of $H$ by $H{\backslash}E^{\prime}$, and for $e\in E$, we write $H{\backslash}e$ instead of $H{\backslash}\{e\}$. For any multiset $E^{\prime}$ of $2^{V}$, the symbol $H+E^{\prime}$ will denote the hypergraph obtained from $H$ by adjoining all edges in $E^{\prime}$.

A $(v_{0},v_{k})$-walk of length $k$ in a hypergraph $H$ is an alternating sequence $W=v_{0}e_{1}v_{1}\ldots$ $v_{k-1}e_{k}v_{k}$ of (possibly repeated) vertices and edges such that $v_{0},\ldots,v_{k}\in V$, $e_{1},\ldots,e_{k}\in E$, and for each $i\in\{1,\ldots,k\}$, the vertices $v_{i-1}$ and $v_{i}$ are adjacent in $H$ via the edge $e_{i}$. Note that since adjacent vertices are by definition distinct, no two consecutive vertices in a walk can be the same. It follows that no walk in a hypergraph contains an edge of cardinality less than 2. The vertices in $V_{a}(W)=\{v_{0},\ldots,v_{k}\}$ are called the anchors of $W$, vertices $v_{0}$ and $v_{k}$ are the endpoints of $W$, and $v_{1},\ldots,v_{k-1}$ are the internal vertices of $W$. We also define the edge set of $W$ as $E(W)=\{e_{1},\ldots,e_{k}\}$, and the set of anchor flags of $W$ as $F(W)=\{(v_{0},e_{1}),(v_{1},e_{1}),(v_{2},e_{2}),\ldots,(v_{k-1},e_{k}),(v_{k},e_{k})\}$. Walks $W$ and $W^{\prime}$ in a hypergraph $H$ are said to be edge-disjoint if $E(W)\cap E(W^{\prime})=\emptyset$, and anchor-disjoint if $V_{a}(W)\cap V_{a}(W^{\prime})=\emptyset$.

A walk $W=v_{0}e_{1}v_{1}\ldots v_{k-1}e_{k}v_{k}$ is called closed if $v_{0}=v_{k}$ and $k\geq 2$; a trail if the edges $e_{1},\ldots,e_{k}$ are pairwise distinct; a path if it is a trail and the vertices $v_{0},\ldots,v_{k}$ are pairwise distinct; and a cycle if it is a closed trail and the vertices $v_{0},\ldots,v_{k-1}$ are pairwise distinct. (Note that in [2], a “trail” was defined as a walk with no repeated anchor flags, and a walk with no repeated edges was called a “strict trail”. In this paper, we shall consider only strict trails, and hence use the shorter term “trail” to mean a “strict trail”.)

A walk $W=v_{0}e_{1}v_{1}\ldots v_{k-1}e_{k}v_{k}$ is said to traverse a vertex $v$ and edge $e$ if $v\in V_{a}(W)$ and $e\in E(W)$, respectively. More specifically, the walk $W$ is said to traverse the edge $e$ via vertex $v$, as well as traverse vertex $v$ via edge $e$, if either $ev$ or $ve$ is a subsequence of $W$.

Vertices $u$ and $v$ are connected in a hypergraph $H$ if there exists a $(u,v)$-walk — or equivalently, a $(u,v)$-path [2, Lemma 3.9] — in $H$, and $H$ itself is connected if every pair of vertices in $V$ are connected in $H$. The connected components of $H$ are the maximal connected subhypergraphs of $H$ without empty edges. The number of connected components of $H$ is denoted by $c(H)$.

An Euler tour of a hypergraph $H$ is a closed trail of $H$ traversing every edge of $H$, and a hypergraph is said to be eulerian if it either admits an Euler tour or is trivial and empty. An Euler family of $H$ is a set of pairwise edge-disjoint and anchor-disjoint closed trails of $H$ jointly traversing every edge of $H$. Clearly, an Euler family of cardinality 1 corresponds to an Euler tour, and vice-versa. An Euler family $\cal F$ of $H$ is said to be spanning if every vertex of $H$ is an anchor of a closed trail in $\cal F$, and an Euler tour of $H$ is spanning if it traverses every vertex of $H$.

The following two observations will be frequently used tools in the rest of the paper. Their easy proofs are omitted.

Lemma 2.2 also gives rise to the following tool, to be used in our algorithms.

As in graphs [3], an edge cut in a hypergraph $H=(V,E)$ is a (multi)set of edges of the form $[S,V-S]_{H}$ for some non-empty proper subset $S$ of $V$. Here, for any $S,T\subseteq V$, we denote

An edge $e$ in a hypergraph $H$ is said to be a cut edge of $H$ if $c(H{\backslash}e)>c(H)$. Thus, an edge $e$ is a cut edge if and only if $\{e\}$ is an edge cut.

Conversely, assume $H$ has an edge cut $F^{\prime}\subseteq F$, where $F^{\prime}=[S,V-S]_{H}$ for $\emptyset\subsetneq S\subsetneq V$. Then $H{\backslash}F$ has no edge intersecting both $S$ and $V-S$, so it is disconnected. a

As we shall see in the next lemma, minimal edge cuts — that is, edge cuts that are not properly contained in other edge cuts — have a very nice property that will be much exploited in subsequent results. Note that, for a hypergraph $H=(V,E)$ and its subhypergraph $H^{\prime}$, we say that an edge $e\in E$ intersects $H^{\prime}$ if $e\cap V(H^{\prime})\neq\emptyset$.

Conversely, assume each edge of $F$ intersects every connected component of $H{\backslash}F$. Suppose $F^{\prime}$ is an edge cut of $H$ and $F^{\prime}\subsetneq F$. Take any $f\in F-F^{\prime}$. Since $f$ intersects every connected component of $H{\backslash}F$, the hypergraph $H+f=H{\backslash}F^{\prime}$ is connected. Hence by Lemma 3.1, the set $F^{\prime}$ contains no edge cut of $H$, a contradiction. Hence the edge cut $F$ is minimal. a

In our algorithms, it will be helpful to find minimal edge cuts containing edges of cardinality at least 3. The proof of the next lemma explains how this can be done.

Let $F_{1}=\left\{\!\!\left\{e\in F:e\cap V(H_{1})\neq\emptyset\right\}\!\!\right\}$, so $e^{\ast}\in F_{1}$. Since vertices of $H_{1}$ are disconnected from vertex $x$ in $H{\backslash}F_{1}$, by Lemma 3.1, there exists a minimal edge cut $F^{\ast}\subseteq F_{1}$.

Let $H_{0}^{\ast},H_{1}^{\ast},\ldots,H_{\ell}^{\ast}$ be the connected components of $H{\backslash}F^{\ast}$. Observe that, without loss of generality, we may assume that $x\in V(H_{0})^{\ast}$, while $H_{1}^{\ast},\ldots,H_{\ell}^{\ast}\in\{H_{1},\ldots,H_{k}\}$.

If $\ell\geq 2$, then by Lemma 3.2, each edge of $F^{\ast}$ has cardinality at least 3. Hence we may assume that $\ell=1$. If $H_{1}^{\ast}=H_{1}$, then $e^{\ast}\in F^{\ast}$, and we are done. Otherwise, each edge in ${\mathcal{F}}^{\ast}$ contains $x$, as well as a vertex in $H_{1}$ and a vertex in $H_{1}^{\ast}$, and hence has cardinality at least 3.

We conclude that $e^{\ast}\in F^{\ast}$ or each edge of $F^{\ast}$ has cardinality at least 3, as claimed. a

We are now ready to present the first of the two main tools that we shall use to study eulerian properties of hypergraphs via their edge cuts; namely, an edge cut assignment $\alpha$ associated with an edge cut $F$ of a hypergraph $H$, which maps each edge of $F$ to an unordered pair of connected components (or, more generally, unions of connected components) of $H{\backslash}F$.

Note that, for a finite set $I$, we denote the set of all unordered pairs of elements in $I$, with repetitions in a pair permitted, by $I^{[2]}$; that is, $I^{[2]}=\{ij:i,j\in I\}$. Thus, $|I^{[2]}|=\frac{1}{2}n(n+1)$.

The usefulness of these concepts will be conveyed in the following three observations, the last of which yields necessary and sufficient conditions for a hypergraph to admit an Euler family (tour) via an edge cut assignment $\alpha$ and the associated hypergraph $H^{\alpha}$.

Moreover, if $T$ is an Euler tour of $H^{\alpha}$, then $T^{\alpha}$ is Euler tour of $G^{\alpha}$. a

We point out that Theorem 4.4 forms the basis for all other, more specific reductions, as well as for the algorithms presented in the rest of this paper.

In this section, we shall be using only standard edge cut assignments; that is, edge cut assignments arising from partitions of the form $\{V(H_{i}):i\in I\}$, where the $H_{i}$, for $i\in I$, are the connected components of $H{\backslash}F$, and $F$ is a minimal edge cut in a hypergraph $H$.

We are now ready to prove our first reduction theorem, Theorem 5.2, which states that a hypergraph $H$ with a minimal edge cut $F$ admits an Euler family if and only if certain subhypergraphs of $H$ obtained from the connected components of $H{\backslash}F$ admit Euler families. The analogous result for Euler tours follows in Theorem 5.3.

To show (i) and (ii), let $V^{\prime}=\bigcup_{j\in J}V(H_{j})$. By Lemma 5.1(i), we have that $H^{\alpha}[V^{\prime}]$ is a union of connected components of $H^{\alpha}$, and $H_{i}$, for each $i\in I-J$, is a connected component of $H^{\alpha}$. Since $H^{\alpha}$ has an Euler family, it follows from Lemma 2.1 that $H^{\alpha}[V^{\prime}]$ has an Euler family, as does $H_{i}$, for each $i\in I-J$.

It remains to show that $H[V^{\prime}]$ has a Euler family. Define a bijection $\varphi:E\left(H^{\alpha}[V^{\prime}]\right)\to E\left(H[V^{\prime}]\right)$ by $\varphi(e^{\alpha})=e\cap V^{\prime}$. Then $e^{\alpha}\subseteq\varphi(e^{\alpha})$ for all $e^{\alpha}\in E\left(H^{\alpha}[V^{\prime}]\right)$, so by Lemma 2.2(i), since $H^{\alpha}[V^{\prime}]$ has an Euler family, so does $H[V^{\prime}]$. Moreover, since $H[V^{\prime}]$ is non-empty, this Euler family is non-empty.

Conversely, assume that there exists $J\subseteq I$ with $1\leq|J|\leq|F|$ such that (i) and (ii) hold. Let $V^{\prime}=\bigcup_{j\in J}V(H_{j})$ and $H^{\prime}=H[V^{\prime}]\cup\bigcup_{i\in I-J}H_{i}$. Then by Lemma 2.1(i), the hypergraph $H^{\prime}$ admits an Euler family.

Observe that $E(H^{\prime})=\bigcup_{i\in I}E(H_{i})\cup\{f\cap V^{\prime}:f\in F\}$. Define a bijection $\varphi:E(H^{\prime})\to E(H)$ by $\varphi(f\cap V^{\prime})=f$ for all $f\in F$, and $\varphi(e)=e$ for all $e\in\bigcup_{i\in I}E(H_{i})$. Hence $e\subseteq\varphi(e)$ for all $e\in E(H^{\prime})$, and since $H^{\prime}$ has an Euler family, by Lemma 2.2(i), the hyperhgraph $H$ has an Euler family as well.

It is easy to see that if the Euler family ${\mathcal{F}}$ of $H$ is spanning, then so are the resulting Euler families of $H[V^{\prime}]$ and $H_{i}$, for all $i\in I-J$, and vice-versa. a

Let $V^{\prime}=\bigcup_{j\in J}V(H_{j})$. By Lemma 5.1(i), we have that $H^{\alpha}[V^{\prime}]$ is a connected component of $H^{\alpha}$, as is $H_{i}$, for each $i\in I-J$. Since $H^{\alpha}$ has an Euler tour and $H^{\alpha}[V^{\prime}]$ is not empty, it follows that $H^{\alpha}[V^{\prime}]$ has an Euler tour, and $H_{i}$, for each $i\in I-J$, is empty.

Conversely, assume that there exists $J\subseteq I$ with $1\leq|J|\leq|F|$ such that (i) and (ii) hold. Let $V^{\prime}=\bigcup_{j\in J}V(H_{j})$, and $H^{\prime}=H[V^{\prime}]\cup\bigcup_{i\in I-J}H_{i}$. Similarly to the proof of Theorem 5.2, we observe that since $H[V^{\prime}]$ has an Euler tour and $H_{i}$, for each $i\in I-J$, is empty, Lemma 2.1(i) shows that $H^{\prime}$ has an Euler tour as well. By Lemma 2.2(i), it follows that $H$ has an Euler tour. a

We observe that Theorem 5.3 cannot be extended to spanning Euler tours in any meaningful way. The following immediate consequence of Theorems 5.2 and 5.3 gives more transparent necessary and sufficient conditions for the existence of an Euler family and Euler tour for a hypergraph with a cut edge.

In Algorithm 5.5 below, we shall now put to use the results of Lemma 4.2 and Theorem 4.4 (applied to standard edge cut assignments), as well as Corollary 2.3 and Lemma 3.3, to determine whether a given hypergraph has an Euler family.

Similarly, in Algorithm 5.6 below, we use Theorem 5.3 and Corollary 5.4, in addition to Corollary 2.3, Lemmas 3.3 and 4.2, and Theorem 4.4, to determine whether a given hypergraph has an Euler tour.