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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| Abstract | 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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| AbstractList | 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]. |
| ArticleNumber | 113825 |
| Author | Yang, Shizhou Munaro, Andrea |
| Author_xml | – sequence: 1 givenname: Andrea orcidid: 0000-0003-1509-8832 surname: Munaro fullname: Munaro, Andrea email: andrea.munaro@unipr.it organization: Department of Mathematical, Physical and Computer Sciences, University of Parma, Italy – sequence: 2 givenname: Shizhou surname: Yang fullname: Yang, Shizhou email: syang22@qub.ac.uk organization: School of Mathematics and Physics, Queen's University Belfast, United Kingdom |
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| Keywords | XP algorithm Mim-width Sim-width Hereditary graph class Width parameter |
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