Bounding the mim‐width of hereditary graph classes

Abstract

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration.We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied.We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H1, H2)-free graphs. We identify several general classes of (H1, H2)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H1, H2)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H1 and H2, the class of (H1, H2)-free graphs has unbounded mim-width.Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H1, H2)-free graphs. In particular, we classify the mim-width of (H1, H2)-free graphs for all pairs (H1, H2) with |V (H1)| + |V (H2)| ≤ 8. When H1 and H2 are connected graphs, we classify all pairs (H1, H2) except for one remaining infinite family and a few isolated cases.

Steiner trees for hereditary graph classes: A treewidth perspective

“…For Open Problem 2 we note that Vertex Steiner Tree is polynomial-time solvable for P 4 -free graphs by Theorem 2. It is known that (K 1,3 , P 5 )-free graphs have unbounded mim-width [7]. Hence, the first open case is where t = 5.…”

“…For the less restrictive width parameter clique-width (or equivalently boolean-width, rank-width, module-width, or NLC-width [8,18,22,24]) or the even less restrictive width parameter mim-width, such dichotomies have not yet been established for (H 1 , H 2 )-free graphs. We refer to [12] and [7] for state-ofthe-art summaries for clique-width and mim-width, respectively.…”

Computing Subset Transversals in $H$-Free Graphs

“…Papadopoulos and Tzimas [32,33] proved that Subset Feedback Vertex Set is polynomial-time solvable for sP 1 -free graphs for any s ≥ 1, cobipartite graphs, interval graphs and permutation graphs, and thus P 4 -free graphs. Some of these results were generalized by Bergougnoux et al [2], who solved an open problem of Jaffke et al [23] by giving an n O(w 2 ) -time algorithm for Subset Feedback Vertex Set given a graph and a decomposition of this graph of mim-width w. This does not lead to new results for H-free graphs: a class of H-free graphs has bounded mim-width if and only if H ⊆ i P 4 [7].…”

Tree pivot-minors and linear rank-width

“…Such a classification already exists for H-free graphs, as observed in [22]: for a graph H, the class of H-free graphs has bounded linear rank-width if and only if H is a subgraph of P 3 not isomorphic to 3P 1 . We note that similar classifications also exist for other width parameters: for the tree-width of pH 1 , H 2 q-free graphs [7], the rank-width of H-free graphs (see [22]), rank-width of H-free bipartite graphs [23,35,36], and up to five non-equivalent open cases, rank-width of pH 1 , H 2 q-free graphs (see [8] or [22]), mim-width of H-free graphs, whereas there is still an infinite number of open cases left for the mim-width of pH 1 , H 2 q-free graphs [13].…”