Tree-width and branch-width are two prominent graph width parameters extensively studied in both structural graph theory and theoretical computer science due to wide applications. For instance, the seminar work by Courcelle  proved that many NP-hard problems in graphs can be solved in polynomial time when the graph has bounded tree-width or branch-width, proving the usefulness the width parameters.
Recently, a new graph width parameter compatible with tree-width or branch-width was introduced, called the maximum matching width (mm-width for short) . It is shown in  that if , , and are the mm-width, tree-width, and branch-width of a graph , respectively, then, for every graph ,
Because of the similarity between the parameters one might anticipate that the new parameter can be replaced by the existing width parameters, questioning the necessity of studying the new parameter. On the other hand, it has been observed that the new parameter can lead to better results than others when employed to design algorithms for popular problems . In particular, it is recently shown that the algorithm for the Minimum Dominating Set problem based on mm-width can enjoy faster time-complexity than one based on tree-width .
In this short paper, we investigate the hardness of computing the mm-width. While it is widely known that the problem of computing the tree-width or branch-width in general graphs are both NP-hard [1, 7], it has not been explored whether or not computing mm-width is NP-hard. We prove that computing the mm-width is also NP-hard by leveraging recent results in .
All graphs are simple, undirected, and finite. For a graph , let and denote the vertex set and edge set of the graph, respectively. For a subset of vertices , is the subset of vertices that are adjacent to at least one vertex in . A tree is called nontrivial if it has at least one edge. We say that a tree is ternary if all vertices have degree or . A function is symmetric if for all , and a function is submodular if for all .
2.1 Maximum matching width
We provide a formal definition of the maximum matching width. A branch-decomposition over a finite set is a pair of a ternary tree together with a bijection from the leaves of to . When we remove an edge of , the tree is divided into two connected components, inducing the partition of due to bijection . Here, we call the partition induced by .
For an edge of , and a function , which is symmetric and submodular, the -value of is where is the partition induced by . The -width of a branch-decomposition is the maximum of -values over . The -width of is the minimum value of the -width over all possible branch-decompositions over .
Let be a function, where is defined as the size of a maximum matching in between and its complement. Note that the function is symmetric and submodular . The maximum matching width of , denoted by , is the -width of .
Jeong, Telle, and Sæther  gave a new characterization of graphs whose mm-width are at most , formally stated as the following theorem.
Theorem 2.1 (Jeong, Telle, and Sæther [4, Theorem 3.8]).
A nontrivial graph has if and only if there exist a ternary tree and nontrivial subtrees of for all vertices such that
if then the subtrees and have at least one node of in common, and
for every edge of there are at most subtrees using this edge.
A tree-representation of a graph is a pair where is a ternary tree and a collection of nontrivial subtrees of satisfying the property . Theorem 2.1 states that a graph G has a tree-representation in which every edge of is contained in at most subtrees if and only if .
2.2 Helly property of subtrees
A set system is said to satisfy the Helly property if the following holds for every subcollection :
It is well known that a collection of the node sets of subtrees of a tree satisfies the Helly property:
Let be a clique of a graph with at least two vertices. For every tree representation , .
3 Computing maximum matching width is NP-hard
A graph is called a split graph if can be partitioned into an independent set of and a clique of , in which case we write and . Inspired by , we prove, in particular, that computing the mm-width of split graphs is NP-hard.
The following lemma characterizes the conditions satisfied by a certain type of split graphs with mm-width equal to :
Let be a split graph with , . Then if and only if can be partitioned into three subsets , , and with such that, for each vertex , is contained exactly one of , , and .
() Let . Assume that is a tree-representation of in which every edge of is contained in at most subtrees, whose existence is ensured by Lemma 2.1. By Proposition 2.2, there exists a vertex in . Then cannot be a leaf; otherwise, the unique edge incident with would be contained in subtrees for all . Hence, the degree of is , and let be the three incident edges in . Let be the number of subtrees containing for every . Then, for every . For each , contains at least one edge among because it contains . Hence, and it follows that for each . This also implies that each tree with contains exactly one edge among . Therefore, by defining , one can partition into with .
Note that the graph obtained from by deleting consists of three disconnected components. Denote by the component that is incident with for each . For each , contains none of because the edges are fully occupied by subtrees corresponding to vertices in . Thus, should be entirely contained in for some . This implies that, for each , if , then . In other words, .
() Assume that admits a partition satisfying the property. Let for every . Moreover, assume that is partitioned into so that for each , . Let for every . For every tree-representation , Proposition 2.2 ensures that there exists a vertex in . As is ternary, is incident with at most three edges, and from the pigeonhole principle, at least one edge should be contained in at least subtrees. Thus, .
Now, we show that . It is enough to construct a tree-representation of where every edge in is contained in at most subtrees by Lemma 2.1. First, introduce many vertices , and . We construct a ternary tree as follows:
Build three paths for every .
Join the three paths by adding three edges for every .
For , add an edge between and . In addition, attach to for every .
In this way, we obtain a ternary tree whose leaves are ’s.
See Figure 1 for an example. Now we define the collection of subtrees as follows:
For each , is the subtree consisting of a single edge .
For each , is the unique path from to .
Then, it is straightforward to check that this construction yields a desired tree-representation. ∎
We now introduce a decision problem, which we call PARTITION-3: Given a multi-set (a set in which multiple elements are allowed) of positive integers, the task is to decide whether can be partitioned into three multi-subsets such that .
For instance, when , the answer is NO as the sum of elements is not even divisible by ; when , the answer is NO as every subset containing the element will exceed a third of the total sum; when , the answer is YES as , , gives a desired partition.
PARTITION-3 is NP-hard.
We construct a polynomial reduction from PARTITION (problem [SP12] in ): The instance of PARTITION is the same as PARTITION-3, and the task is to decide whether a multiset can be partitioned into two multi-subsets of equal sums rather than three. The reduction is constructed as follows: for a given instance of PARTITION, construct an instance of PARTITION as . The correctness of this reduction is straightforward. Because PARTITION is known to be NP-hard , PARTITION-3 is NP-hard. ∎
Now, we finish the proof.
Computing the maximum matching width for graphs is NP-hard. In particular, computing the maximum matching width for split graphs is NP-hard.
The reduction is from PARTITION-3. For a given instance , we construct a split graph as follows:
Consider a complete graph on vertices . Partition into many subsets so that for each .
Introduce more vertices , and connect to all vertices in for each .
By Lemma 3.1, such constructed split graph has mm-width equal to if and only if can be partitioned into three equal-sized partition such that for each , is completely contained in one of three partitions. By letting for every , one can see that . ∎
We thank Jan Arne Telle, Sigve Hortemo Sæther, and Yixin Cao for fruitful advice.
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