Bounding the Mim-Width of Hereditary Graph Classes
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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.
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