Width parameters play an important role in algorithmic graph theory, as evidenced by various surveys [12, 18, 19, 30, 31]. A graph class has bounded width, for some width parameter, if there exists a constant such that every graph in has width at most . Mim-width is a relatively young width parameter that was introduced by Vatshelle . It is defined as follows. A branch decomposition for a graph is a pair , where is a subcubic tree and is a bijection from to the leaves of . Every edge partitions the leaves of into two classes, and , depending on which component of they belong to. Hence, induces a partition of , where and . We let be the bipartite subgraph of induced by the edges with one end-vertex in and the other in . A matching of is induced if there is no edge in between vertices of different edges of . We let be the size of a maximum induced matching in . The mim-width of is the maximum value of over all edges . The mim-width of is the minimum value of over all branch decompositions for .
Vatshelle  proved that every class of bounded clique-width, or equivalently, bounded boolean-width, module-width, NLC-width or rank-width, has bounded mim-width, and that the converse is not true. That is, he proved that there exist graph classes of bounded mim-width that have unbounded clique-width. This means that proving that a problem is polynomial-time solvable for graph classes of bounded mim-width (see [1, 2, 3, 7, 13, 21, 22, 23] for examples of such problems) yields more tractable graph classes than doing this for clique-width. Hence, mim-width has greater modeling power than clique-width. However, the trade-off is that fewer problems admit such an algorithm, as we explain below by means of a relevant example, namely the classical Colouring problem. Moreover, computing mim-width is NP-hard  and, in contrast to the situation of clique-width , it remains a challenging open problem to develop a polynomial-time algorithm for approximating the mim-width of a graph (doing this within a constant factor of the optimal is not possible unless ). This has the following implication. For a class of graphs with bounded mim-width, we need an explicit polynomial-time algorithm for computing a branch decomposition whose mim-width is bounded by a constant. When this is possible, we say that the mim-width of is quickly computable. We can then develop a polynomial-time algorithm for the problem of interest via dynamic programming over the computed branch decomposition; see [5, 7, 24] for some examples.
A colouring of a graph is a mapping that gives each vertex a colour in such a way that, for every two adjacent vertices and , we have that . If for every we have , then we say that is a -colouring of . The Colouring problem is to decide whether a given graph has a -colouring for some given integer . If is fixed, that is, not part of the input, we call this the -Colouring problem. A classical result of Lovász  states that -Colouring is NP-complete even if .
The Colouring problem is an example of a problem that distinguishes between classes of bounded mim-width and bounded clique-width: it is polynomial-time solvable for every graph class of bounded clique-width  but NP-complete for circular-arc graphs , which form a class whose mim-width is bounded and quickly computable . When we fix , we no longer have this distinction, as -Colouring, for every fixed integer , is polynomial-time solvable for a graph class whose mim-width is bounded and quickly computable . Kwon  observed that an instance of the more general List -Colouring problem (defined below) can be reduced to an instance of -Colouring while only increasing the mim-width of the input graph by at most (see also ). Hence, the List -Colouring problem is also polynomial-time solvable for a graph class whose mim-width is bounded and quickly computable.
For an integer , a -list assignment of a graph is a function that assigns each vertex a list of admissible colours for . A colouring of respects if for every . For a fixed integer , the List -Colouring problem is to decide whether a given graph with a -list assignment admits a colouring that respects . Note that for , List -Colouring is a special case of List -Colouring.
[] For every , List -Colouring is polynomial-time solvable for a graph class whose mim-width is bounded and quickly computable.
A graph is -free, for some graph , if it contains no induced subgraph isomorphic to . For a set of graphs , a graph is -free if it is -free for every . We denote the disjoint union of two graphs and by . We let and denote the path and complete graph on vertices, respectively.
The complexity of Colouring for -free graphs has been settled for every graph , but there are still infinitely many open cases for -Colouring restricted to -free graphs when is a linear forest, that is, a disjoint union of paths. We refer to  for a survey and to [8, 10, 25] for updated summaries. In particular, Hoàng et al.  proved that for every integer , -Colouring is polynomial-time solvable for -free graphs. This result was generalized by Couturier et al.  as follows:
[] For every and , List -Colouring is polynomial-time solvable for -free graphs.
We may assume without loss of generality that an instance of List -Colouring is -free, for otherwise it is a no-instance. Hence, Theorem 1 also immediately follows from combining Theorem 1 with the following recent result.
[] For every and , the mim-width of the class of -free graphs is bounded and quickly computable.
Theorem 1 is part of a recent study [6, 5] on the boundedness of mim-width for hereditary graph classes characterized by a small set of forbidden induced subgraphs. If , then has bounded mim-width if and only if has bounded clique-width . On the other hand, this equivalence does not always hold when .
For and , we let denote the complete bipartite graph with partition classes of size and . The graph is also known as the -vertex star. The -subdivision of a graph is the graph obtained from by subdividing each edge of exactly once. We denote the -subdivision of a star by ; in particular .
Even more recently than , Chudnovsky, Spirkl and Zhong proved the following result.
[] For every , and , List -Colouring is polynomial-time solvable for -free graphs.
For every and , the class of -free graphs contains the class of -free graphs. Hence, Theorem 1 generalizes Theorem 1 for the case where . As is an induced subgraph of , Theorem 1 also generalizes the following result for the case where .
[] For every , , and , List -Colouring is polynomial-time solvable for -free graphs.
In this note we generalize Theorem 1.
For every , and , the mim-width of the class of -free graphs is bounded and quickly computable.
For every , and , List -Colouring is polynomial-time solvable for -free graphs.
Corollary 1 is tight in the following sense. Let denote the subgraph obtained from by subdividing one edge exactly once; in particular . Then, as List -Colouring is NP-complete for -free graphs , we cannot generalize Corollary 1 to -free graphs for , and . Moreover, the mim-width of -free graphs is unbounded  and so we cannot extend Theorem 1 to -free graphs, for , and , either.
The List -Colouring problem is polynomial-time solvable for -free graphs , but the computational complexities of -Colouring and List -Colouring is open for -free graphs if . In particular, we do not know any integer such that -Colouring or List -Colouring are NP-complete for -free graphs. Hence, an extension of Corollary 1 might still be possible for , and we leave this for future work. This requires more research into the structure of -free graphs.
Theorem 1 has other applications as well. For a graph , let denote the size of a maximum clique in . Chudnovsky et al.  gave for the class of -free graphs, an -time algorithm for Max Partial -Colouring, a problem equivalent to Independent Set if and to Odd Cycle Transversal if . In other words, Max Partial -Colouring is polynomial-time solvable for -free graphs with bounded clique number. Chudnovsky et al.  noted that Max Partial -Colouring is polynomial-time solvable for graph classes whose mim-width is bounded and quickly computable. Hence, Theorem 1 generalizes their result for Max Partial -Colouring to -free graphs with bounded clique number, for any and . However, the running time of the corresponding algorithm is worse than (see  for details).
It remains to prove Theorem 1, which we do in the next section.
2 The Proof of Theorem 1
We first state two lemmas. The first lemma shows that given a partition of the vertex set of a graph , we can bound the mim-width of in terms of the mim-width of the graphs induced by each part and the mim-width between any two of the parts.
[] Let be a graph, and let be a partition of such that for all distinct , and . Then
Moreover, if is a branch decomposition of for each , then we can construct, in time, a branch decomposition of with .
A clique in a graph is a set of pairwise adjacent vertices. An independent set is a set of pairwise non-adjacent vertices. A dominating set is a set of vertices such that every vertex not in is adjacent to at least one vertex in . For the second lemma we need Ramsey’s Theorem. This theorem states that for every two positive integers and , there exists an integer such that every graph on at least vertices contains a clique of size or an independent set of size .
For and , let . The next lemma generalizes a result of Chudnovsky, Spirkl and Zhong  for the case where to . The proof of this generalization is analogous to the proof in  for the case where (replace each occurrence of “” in the proofs of Lemma 11 and 13 in  by “”).
For every , and , a connected -free graph contains a dominating set of size at most .
Theorem 1 (restated). For every , and , the mim-width of the class of -free graphs is bounded and quickly computable.
Let be a -free graph for some , and . We may assume without loss of generality that is connected. We use induction on . If , then the statement of the theorem holds trivially.
Suppose that . By Lemma 2, we find that has a dominating set of size at most , which is a constant, as and are fixed. Moreover, we can find in polynomial time by brute force (or we can apply the -time algorithm of ). We let .
We will partition the vertex set of with respect to as follows. We first fix an arbitrary ordering on the vertices of . Let be the set of vertices in adjacent to . For , let be the set of vertices in adjacent to , but non-adjacent to any with . Then and the sets partition (some of the sets might be empty).
By construction, is adjacent to every vertex of for each . As is -free, this implies that each induces a -free subgraph of . We denote this subgraph by . By the induction hypothesis, the mim-width of is bounded and quickly computable for every .
Consider two sets and with . We claim that . Towards a contradiction, suppose that . Then, by definition, there exist two sets and , each of size , such that is a set of edges with the property that does not contain any edges for (note that edges and may exist in ).
As is -free, Ramsey’s Theorem tells us that the subgraph of induced by contains an independent set of size . Assume without loss of generality that . Let . As is -free, the subgraph of induced by contains an independent set of size . Assume without loss of generality that . By construction, is adjacent to every vertex of and non-adjacent to every vertex of . Hence, induces a in , a contradiction. We conclude that .
We now apply Lemma 2 to find that the mim-width of is bounded and quickly computable. Let be a branch decomposition for with mim-width . We can readily extend to a branch decomposition for with mim-width at most . Namely, we can obtain from by identifying one leaf of with a leaf of an arbitrary subcubic tree with leaves. Recall that we found the set in polynomial time. We conclude that the mim-width of is bounded and quickly computable. ∎
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