Bounding the mim-width of hereditary graph classes

A large number of (Formula presented.) -hard graph problems are solvable in (Formula presented.) 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 maximum-induced matching width (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 (Formula presented.), the class of (Formula presented.) -free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (Formula presented.) -free graphs. We identify several general classes of (Formula presented.) -free graphs having unbounded clique-width, but bounded mim-width; moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time for these classes. Hence, these results have algorithmic implications: when the input is restricted to such a class of (Formula presented.) -free graphs, many problems become polynomial-time solvable, including classical problems, such as (Formula presented.) - Colouring and Independent Set, domination-type problems known as Locally Checkable Vertex Subset and Vertex Partitioning (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 (Formula presented.) and (Formula presented.), the class of (Formula presented.) -free graphs has unbounded mim-width. Boundedness of clique-width implies boundedness of mim-width. By combining our results 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 (Formula presented.) -free graphs. In particular, we classify the mim-width of (Formula presented.) -free graphs for all pairs (Formula presented.) with (Formula presented.). When (Formula presented.) and (Formula presented.) are connected graphs, we classify all pairs (Formula presented.) except for one remaining infinite family and a few isolated cases.