On the time complexity of finding a well-spread perfect matching in bridgeless cubic graphs^††thanks: Supported by the project GAUK182623 of the Charles University Grant Agency and by grant 25-16627SS of the Czech Science Foundation.

It is well known that every bridgeless cubic graph admits a perfect matching [Pet91]. Using Edmonds’ blossom algorithm, a maximum matching can be found in polynomial time. Gabow [Gab90] demonstrated that the weighted matching problem on general graphs can be solved in time $O(n(m+n\log n))$.

Consequently, the minimum weight perfect matching problem for bridgeless cubic graphs can be solved in $O(n^{2}\log n)$ time. Diks and Stańczyk [DS10] proposed an improved algorithm for finding perfect matchings in bridgeless cubic graphs with time complexity of $O(n\log^{2}n)$.

In this paper we study the complexity of finding a perfect matching $M$ in a bridgeless cubic graph such that in every 3-edge cut $M$ contains exactly one edge. By parity, $M$ can contain one or three edges in a 3-edge cut; so our condition says no 3-edge cut is contained in $M$. We shortly express this by saying $M$ is well spread.

Kaiser et al. [KKN06] showed that a wells spread perfect matching exists in every bridgeless cubic graph (and went on to use this to find how much of the graph can be covered by two, three, etc. perfect matchings). Their proof uses Edmonds perfect matching polytope (vector $(1/3,1/3,\dots)$ is a fractional perfect matching, thus it is a convex combination of perfect matchings – each of them is well spread). This, however, doesn’t yield an efficient algorithm.

In [BIT13], Boyd et al. focus on developing algorithms for finding $2$-factors in bridgeless cubic graphs that cover specific edge-cuts, which brings these $2$-factors closer to Hamiltonian cycles. They provide an efficient algorithm that finds a minimum-weight $2$-factor that covers all $3$-edge cuts in weighted bridgeless cubic graphs (in contrast with finding a Hamiltonian cycle, which is computationally hard). They also provide both a polyhedral description of such $2$-factors and of their complements – well spread perfect matchings. This is the first known polynomial-time algorithm for this problem, with a time complexity of $O(n^{3})$, where $n$ is the number of vertices. In this work, we improve this result for $3$-edge-connected cubic graphs using Algorithm 1 with time complexity $O(n\log^{4}n)$.

Boyd et al. find a “peripheral 3-edge-cut” (a cut such that one side of it is internally 4-edge-connected) and then use recursion. We save computation by using the cactus model, also called a tree of cuts. In this tree it is easy to find the peripheral cut and also to update the tree when we contract a part of the cut. This leads to much improved time complexity, although only for 3-edge-connected graphs.

We then proceed to apply this result to the study of the well-known Cycle Double Cover (or CDC) Conjecture [Sze73, Sey79]. In the language of graph embedding, CDC conjecture is equivalent to every bridgeless cubic graph having a surface embedding with no singular edges (i.e., with no edge that is on the boundary of one face twice). As an application of our work, for a $3$-edge-connected cubic graph we can find embeddings with a bounded number of singular edges in time $O(n\log^{4}n)$.

The structure of the paper is as follows. In the next section, we introduce the necessary definitions and concepts. In Section 3, we discuss the cactus model, a key component in deriving our results. Section 4 presents our main result, an efficient algorithm to find the well-spread perfect matching. Finally, in Section 5, we apply our algorithm to get results about the well-known CDC conjecture.

Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. A graph in which all vertices have degree $3$ is called cubic. A cycle is a connected $2$-regular graph. A bridge in a graph $G$ is an edge whose removal increases the number of components of $G$. Equivalently, a bridge is an edge that is not contained in any cycles of $G$. A graph is bridgeless if it contains no bridge. An edge cut in a graph is a set of edges whose removal increases the number of connected components of the graph. An edge cut $C$ in $G$ is called a non-trivial cut if every component of $G-C$ has at least two vertices. Otherwise it is called trivial. A $k$-edge-cut in a graph is an edge cut that contains exactly $k$ edges. A connected graph is $k$-edge-connected if it remains connected whenever fewer than $k$ edges are removed. A graph is cyclically $k$-edge-connected, if at least $k$ edges must be removed to disconnect it into two components such that each component contains a cycle. We say that a subset $F\subset E$ covers an edge-cut $D$ if $F\cap D\neq\emptyset$. For a subset $S\subseteq E$, $G/S$ is the graph obtained from $G$ by contracting all the edges in $S$. Note that we keep multiple edges and only remove loops in such contraction.

Petersen’s theorem [Pet91] states that every bridgeless cubic graph contains a perfect matching. Schönberger [Sch35] proved the following strengthened form of Petersen’s theorem.

A perfect matching containing a specified edge $e^{*}$ in a bridgeless cubic graph can be found in $O(n\log^{4}n)$ time [BBDL01].

A graph $G$ is embedded in a surface $S$ if the vertices of $G$ are distinct elements of $S$ and every edge of $G$ is a simple arc connecting in $S$ the two vertices which it joins in $G$, such that its interior is disjoint from other edges and vertices. An embedding of a graph $G$ in $S$ is an isomorphism of $G$ with a graph $G^{\prime}$ embedded in $S$.

Let $G$ be a graph that is cellularly embedded in a surface $S$, that is, every face is homeomorphic to an open disk. Let $\pi=\{\pi_{v}|v\in V(G)\}$ where $\pi_{v}$ is the cyclic permutation of the edges incident with the vertex $v$ such that $\pi_{v}(e)$ is the successor of $e$ in the clockwise ordering around $v$. The cyclic permutation $\pi_{v}$ is called the local rotation at $v$, and the set $\pi$ is the rotation system of the given embedding of $G$ in $S$.

Let $G$ be a connected multigraph. A combinatorial embedding of $G$ is a pair $(\pi,\lambda)$ where $\pi=\{\pi_{v}|v\in V(G)\}$ is a rotation system, and $\lambda$ is a signature mapping which assigns to each edge $e\in E(G)$ a sign $\lambda(e)\in\{-1,1\}$. If $e$ is an edge incident with $v\in V(G)$, then the cyclic sequence $e,\pi_{v}(e),\pi_{v}^{2}(e),\dots$ is called the $\pi$-clockwise ordering around $v$ (or the local rotation at $v$). Given an embedding $(\pi,\lambda)$ of $G$ we say that $G$ is $(\pi,\lambda)$-embedded. It is known that the combinatorial embedding uniquely determines a cellular embedding to some surface, up to homeomorphism [MT01, Theorems 3.2.4 and 3.3.1].

A closed walk is a sequence $(v_{0},e_{0},v_{1},$ $e_{1},\dots,e_{n-1},v_{n})$ where $v_{0}=v_{n}$ and for every $i$, $e_{i}=\{v_{i},v_{i+1}\}$. We say that a collection of closed walks $C_{1},\dots,C_{n}$ forms a partial circuit double cover (or partial CDC) if each edge is covered at most once by one of $C_{i}$’s or exactly twice by two different $C_{i}$ and $C_{j}$, and for every vertex $v$ and edges $e$ and $f$ where $e\cap f=\{v\}$, there exists at most one closed walk $C_{i}$ such that $\{e,f\}\subseteq E(C_{i})$.

Here we follow the notations and definition of [NI08]. A general way to represent a subset of cuts within a graph $G$ involves constructing a cactus representation $(T,\varphi)$. In this representation, $T$ is a graph, and $\varphi$ is a function mapping vertices from $V(G)$ to $V(T)$. The cactus representation (or the cactus model) was introduced in [DKL76]. It was further utilized in [Gab91, Din93]. This model plays a crucial role in understanding both edge connectivity and graph rigidity problems.

In this context, we use vertex when referring to elements of $V(G)$ and node for elements of $V(T)$. The set $V(T)$ may include a node $x$ that does not correspond to any vertex $v\in V(G)$ with $\varphi(v)=x$; such a node is referred to as an empty node. We use $C(T)$ for the set of all minimum cuts in $T$.

It was shown in [DKL76] that every $G$ has a cactus representation using a cactus graph – a multigraph where every edge is exactly one cycle. A special case of such graph is a tree with every edge doubled. It is known that when the connectivity is odd, the cactus representation is of this special type. In this case we will just call it a tree of cuts (or cactus tree). For a $3$-edge-connected cubic graph $G$, the vertices of the graph $G$ correspond to the leaf nodes of the cactus tree $T$ and all the interior nodes of $T$ are empty nodes (see Fig. 1). We will also use the following estimate for the size of $T$.

In this section, we present an algorithm that finds a perfect matching in a $3$-edge-connected cubic graph that intersects all $3$-edge cuts in exactly one edge and is substantially faster than the algorithm introduced by Boyd et al. [BIT13]. We first explain the underlying technique. Let $G$ be a graph with a 3-edge cut $\{A,{\bar{A}}\}$; that is $|\delta(A)|=3$. We let $G_{1}=G/A$ (all vertices in $A$ are contracted to a single vertex $a$) and $G_{2}=G/{\bar{A}}$ (with a new vertex ${\bar{a}}$). A well-spread matching $M$ in $G$ contains exactly one edge of $\delta(A)$, thus it gives us a perfect matching $M_{1}$ in $G_{1}$ and $M_{2}$ in $G_{2}$. As edges out of $a$ in $G_{1}$ and out of ${\bar{a}}$ in $G_{2}$ are in natural correspondence with edges of $\delta(A)$, we will identify them and say that $M_{1}$ and $M_{2}$ agree on $\delta(A)$. For our algorithm the following is crucial.

Boyd et al. [BIT13] use the above theorem together with repeated finding of a “peripheral 3-edge-cut” $\{A,{\bar{A}}\}$: one where $G/A$ is internally 4-edge-connected. We refine their approach by utilizing (and updating) the model for all 3-edge cuts that will allow us to find peripheral cuts quickly. Also, we will use the model of the cuts to quickly decompose the graph $G$ in $G/A$ and $G/{\bar{A}}$ and then combine them back.

We will also use the following information about structure of cuts in $G_{1}$ and $G_{2}$.

The proof follows the same idea as that of Theorem 4.1: minimum odd edge-cuts do not cross. We omit the details in this extended abstract.

In our application we have a 3-regular 3-edge-connected graph, so all vertices are mapped to leaf nodes of the cactus tree and nontrivial 3-cuts correspond to edges that are not incident with a leaf. In particular, when the cactus tree is a star there are no nontrivial 3-cuts and we may simply find any perfect matching. In a sense, the purpose of our algorithm is to efficiently combine matchings coming from these “special cases” of internally 4-edge-connected graphs.

To this end, we pick root in the cut model $T$. Let $e_{x}$ be the edge connecting $x$ to its parent. We let $C_{x}$ be the three edges in the cut corresponding to $e_{x}$. We let $A_{x}$ be the vertices of $G$ that are mapped by $\varphi$ to descendants of $x$ (including $x$), so that $C_{x}=\delta(A_{x})$.

We do not use this definition directly, instead we recursively update $A_{x}$, $C_{x}$ while traversing the tree. We can compute $A_{x}$ as a union of $A_{y}$ for all children $y$ of $x$. We can compute $C_{x}$ by going over $C_{y}$ for all children $y$ of $x$, excluding edges that lead withing $A_{x}$ (between $A_{y}$ and $A_{y^{\prime}}$ for two children of $x$). To do this efficiently we utilize the Union-Find algorithm.

In a typical step of decomposition part of the algorithm, we use Theorem 4.2, with the new graphs being $G_{x}$ (which we store for later) and updated version of $G$. To illustrate the algorithm, in Figure 2 we show one step including the updated trees. We see that $T_{x}$ is a star – which means $G_{x}$ is internally 4-edge-connected. Updated $T$ simply removes children of $x$. However, we actually do not need to update $T$ and compute $T_{x}$ in the algorithm.

In a typical step of assembling part of the algorithm, we use Theorem 4.1. For graph $G$ we have our well-spread matching from the recursion. For the other graph, $G_{x}$, we compute it easily, as the graph is internally 4-edge-connected, so any perfect matching will do. Figure 3 illustrates this part of the algorithm.

In [GŠ24], Ghanbari and Šamal establish an upper bound of $\frac{n}{10}$ on the number of singular edges in an embedding of a bridgeless cubic graph on a surface. They also raise the question of how efficiently one can find a perfect matching in a bridgeless cubic graph that contains no odd cut of size $3$ — an essential step in determining the time complexity of finding such an embedding. Here, we answer their question for $3$-edge connected cubic graphs and describe an algorithm that constructs an embedding with at most $\frac{n}{10}$ bad edges. To provide context, we first state their results.

To prove Theorem 5.1, in [GŠ24], we first construct a perfect matching $M_{1}$ such that $M_{1}$ contains no odd cut of size 3. This guarantees the existence of another perfect matching $M_{2}$ satisfying $|M_{1}\cap M_{2}|\leq\frac{n}{10}$ (see [KKN06]). Next, we show that $M_{1}\cap M_{2}$ can be extended to an embedding using Lemma 2, where the potential singular edges of the embedding are precisely those in $M_{1}\cap M_{2}$, and this completes their proof.

Suppose that the cubic graph $G$ is $3$-edge-connected. To find $M_{1}$, we employ Algorithm 1, which finds a perfect matching in time $O(n\log^{4}n)$. Consequently, determining the overall time complexity reduces to finding the perfect matching $M_{2}$. We utilize Diks and Stańczyk’s algorithm [DS10] to find $M_{2}$ in time $O(n\log^{2}n)$. Thus, the embedding guaranteed by Theorem 5.1 can be found in total time

We studied a new approach to finding a perfect matching that includes all $3$-edge cuts in a $3$-edge-connected cubic graph, utilizing the cactus representation of the graph. We referred to such a perfect matching as well spread. In Section 4, we presented an algorithm that finds a well-spread perfect matching in a $3$-edge-connected cubic graph in time $O(n\log^{4}n)$.

In general, the best-known algorithm for finding a well-spread perfect matching in a bridgeless cubic graph has a time complexity of $O(n^{3})$ [BIT13]. In the future, it would be interesting to determine whether the cactus representation can be leveraged to develop a faster algorithm for finding such a well-spread perfect matching. However, the issue is that representing 3-edge-cuts in a graph that may contain 2-edge-cuts is more complicated.