## 1 Introduction

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 [34].
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 [34] 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 [33] and, in contrast to the situation of clique-width [32], 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 [33]). 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 [29] 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 [26] but NP-complete for circular-arc graphs [14], which form a class whose mim-width is bounded and quickly computable [1]. 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 [7]. Kwon [28] 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 [6]). 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.

[[28]] For every , List -Colouring is polynomial-time solvable for a graph class whose mim-width is bounded and quickly computable.

In this note, we continue previous work [6] and show that known polynomial-time results for List -Colouring on special graph classes can be obtained, and strengthened, by applying Theorem 1.

### Related Work

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 [27], 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 [15] for a survey and to [8, 10, 25] for updated summaries. In particular, Hoàng et al. [20] proved that for every integer , -Colouring is polynomial-time solvable for -free graphs. This result was generalized by Couturier et al. [11] as follows:

[[11]] 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.

[[6]] 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 [5]. On the other hand, this equivalence does not always hold when [5].

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 [6], Chudnovsky, Spirkl and Zhong proved the following result.

[[10]] 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 .

[[17]] For every , , and , List -Colouring is polynomial-time solvable for -free graphs.

### Our Result

In this note we generalize Theorem 1.

For every , and , the mim-width of the class of -free graphs is bounded and quickly computable.

Recall that an instance of List -Colouring may be assumed to be -free. Combining Theorem 1 with Theorem 1 enables us to generalize both Theorems 1 and 1.

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 [16], we cannot generalize Corollary 1 to -free graphs for , and . Moreover, the mim-width of -free graphs is unbounded [5] and so we cannot extend Theorem 1 to -free graphs, for , and , either.

The List -Colouring problem is polynomial-time solvable for -free graphs [4], 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. [9] 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. [9] 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 [9] 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.

[[6]] 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 [10] for the case where to . The proof of this generalization is analogous to the proof in [10] for the case where (replace each occurrence of “” in the proofs of Lemma 11 and 13 in [10] by “”).

For every , and , a connected -free graph contains a dominating set of size at most .

We are now ready to prove Theorem 1, which we restate below. Its proof closely resembles the proof of Theorem 1. Note that the constant bound depends on , and .

Theorem 1 (restated). For every , and , the mim-width of the class of -free graphs is bounded and quickly computable.

###### Proof.

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 [10]). 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. ∎

## References

- [1] Rémy Belmonte and Martin Vatshelle. Graph classes with structured neighborhoods and algorithmic applications. Theoretical Computer Science, 511:54–65, 2013.
- [2] Benjamin Bergougnoux and Mamadou Moustapha Kanté. More applications of the -neighbor equivalence: Connectivity and acyclicity constraints. Proc. ESA 2019, LIPIcs, 144:17:1–17:14, 2019.
- [3] Benjamin Bergougnoux, Charis Papadopoulos, and Jan Arne Telle. Node multiway cut and subset feedback vertex set on graphs of bounded mim-width. Proc. WG 2020, LNCS, to appear.
- [4] Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Oliver Schaudt, Maya Stein, Mingxian Zhong. Three-coloring and List Three-Coloring of graphs without induced paths on seven vertices. Combinatorica, 38:779–801, 2018.
- [5] Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, and Daniël Paulusma. Bounding the mim-width of hereditary graph classes. CoRR, arXiv:2004.05018, 2020.
- [6] Nick Brettell, Jake Horsfield and D. Paulusma, Colouring -free graphs: a mim-width perspective. CoRR, arXiv:2004.05022, 2020.
- [7] Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoretical Computer Science, 511:66–76, 2013.
- [8] Maria Chudnovsky, Shenwei Huang, Sophie Spirkl, and Mingxian Zhong. List-three-coloring graphs with no induced . CoRR, arXiv:1806.11196, 2018.
- [9] Maria Chudnovsky, Jason King, Michał Pilipczuk, Paweł Rza̧żewski, and Sophie Spirkl. Finding large H-colorable subgraphs in hereditary graph classes, Proc. ESA 2020, LIPIcs, to appear.
- [10] Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. List-three-coloring -free graphs with no induced -subdivision of . CoRR, arXiv:2006.03009, 2020.
- [11] Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, and Daniël Paulusma. List coloring in the absence of a linear forest. Algorithmica, 71:21–35, 2015.
- [12] Konrad K. Dabrowski, Matthew Johnson and Daniël Paulusma. Clique-width for hereditary graph classes. London Mathematical Society Lecture Note Series. 456:1-56, 2019.
- [13] Esther Galby, Andrea Munaro, and Bernard Ries. Semitotal Domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width. Theoretical Computer Science, 814:28–48, 2020.
- [14] M. R. Garey, David S. Johnson, G. L. Miller, and Christos H. Papadimitriou. The complexity of coloring circular arcs and chords. SIAM Journal on Matrix Analysis and Applications, 1(2):216–227, 1980.
- [15] Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of colouring graphs with forbidden subgraphs. Journal of Graph Theory, 84:331–363, 2017.
- [16] Petr A. Golovach, Daniël Paulusma, and Jian Song. Closing complexity gaps for coloring problems on -free graphs. Information and Computation 237:204–214, 2014.
- [17] Petr A. Golovach, Daniël Paulusma and Jian Song. Coloring graphs without short cycles and long induced paths. Discrete Applied Mathematics 167:107–120, 2014.
- [18] Frank Gurski. The behavior of clique-width under graph operations and graph transformations. Theory of Computing Systems, 60:346–376, 2017.
- [19] Petr Hliněný, Sang-il Oum, Detlef Seese, and Georg Gottlob. Width parameters beyond tree-width and their applications. The Computer Journal, 51:326–362, 2008.
- [20] Chính T. Hoàng, Marcin Kamiński, Vadim V. Lozin, Joe Sawada, and Xiao Shu. Deciding -Colorability of -free graphs in polynomial time. Algorithmica, 57:74–81, 2010.
- [21] Lars Jaffke, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. Mim-width III. Graph powers and generalized distance domination problems. Theoretical Computer Science, 796:216–236, 2019.
- [22] Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width I. Induced path problems. Discrete Applied Mathematics, 278:153–168, 2020.
- [23] Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width II. The feedback vertex set problem. Algorithmica, 82:118–145, 2020.
- [24] Dong Yeap Kang, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. A width parameter useful for chordal and co-comparability graphs. Theoretical Computer Science, 704:1–17, 2017.
- [25] Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná, Daniël Paulusma, and Veronika Slívová. Colouring -free graphs. Algorithmica, 82:1833–1858, 2020.
- [26] Daniel Kobler and Udi Rotics. Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics, 126:197–221, 2003.
- [27] Daniel Král’, Jan Kratochvíl, Zsolt Tuza, and Gerhard J. Woeginger. Complexity of coloring graphs without forbidden induced subgraphs. Proc. WG 2001, LNCS, 2204:254–262, 2001.
- [28] O. Kwon. Personal communication, 2020.
- [29] László Lovász. Coverings and coloring of hypergraphs. Congressus Numerantium, VIII:3–12, 1973.
- [30] Marcin Kamiński, Vadim V. Lozin, and Martin Milanič. Recent developments on graphs of bounded clique-width. Discrete Applied Mathematics 157:2747–2761, 2009.
- [31] Sang-il Oum. Rank-width: Algorithmic and structural results. Discrete Applied Mathematics 231:15–24, 2017
- [32] Sang-il Oum and Paul Seymour, Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B, 96:514–528, 2006.
- [33] Sigve Hortemo Sæther and Martin Vatshelle. Hardness of computing width parameters based on branch decompositions over the vertex set. Theoretical Computer Science, 615:120–125, 2016.
- [34] Martin Vatshelle. New Width Parameters of Graphs. PhD thesis, University of Bergen, 2012.