On algorithmic applications of sim-width and mim-width of (H1,H2)-free graphs
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 alg...
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| Published in: | Theoretical computer science Vol. 955; p. 113825 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
26.04.2023
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| Subjects: | |
| ISSN: | 0304-3975 |
| Online Access: | Get full text |
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| Summary: | 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. (2017) [32], 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, Listk-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 IndependentH-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 (H1,H2)-free graphs in the case H1 is complete or edgeless. Our results solve some open problems in Brettell et al. (2022) [6]. |
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| ISSN: | 0304-3975 |
| DOI: | 10.1016/j.tcs.2023.113825 |