On algorithmic applications of sim-width and mim-width of (H_1, H_2)-free graphs
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Mim-width and sim-width are among the most powerful graph width parameters, with sim-width more powerful than mim-width, which is in turn more powerful than clique-width. While several 𝖭𝖯-hard graph problems become tractable for graph classes whose mim-width is bounded and quickly computable, no algorithmic applications of boundedness of sim-width are known. In [Kang et al., A width parameter useful for chordal and co-comparability graphs, Theoretical Computer Science, 704:1-17, 2017], it is asked whether Independent Set and 3-Colouring are 𝖭𝖯-complete on graphs of sim-width at most 1. We observe that, for each k ∈ℕ, List k-Colouring is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. Moreover, we show that if the same holds for Independent Set, then Independent ℋ-Packing is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. This problem is a common generalisation of Independent Set, Induced Matching, Dissociation Set and k-Separator. We also make progress toward classifying the mim-width of (H_1,H_2)-free graphs in the case H_1 is complete or edgeless. Our results solve some open problems in [Brettell et al., Bounding the mim-width of hereditary graph classes, Journal of Graph Theory, 99(1):117-151, 2022].
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