Bounding the Mim-Width of Hereditary Graph Classes

Abstract

A large number of NP-hard graph problems become polynomial-time solvable on graph classes where the mim-width is bounded and quickly computable. Hence, when solving such problems on special graph classes, it is helpful to know whether the graph class under consideration has bounded mim-width. We first extend the toolkit for proving (un)boundedness of mim-width of graph classes. This enables us to initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes. 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 $(H_1,H_2)$-free graphs. We find several general classes of $(H_1,H_2)$-free graphs having unbounded clique-width, but the mim-width is bounded and quickly computable. We also prove a number of new results showing that, for certain $H_1$ and $H_2$, the class of $(H_1,H_2)$-free graphs has unbounded mim-width. Combining these with known results, we present summary theorems of the current state of the art for the boundedness of mim-width for $(H_1,H_2)$-free graphs.