Dynamic programming on bipartite tree decompositions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917108711948288 |
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| author | Jaffke, Lars Morelle, Laure Sau, Ignasi Thilikos, Dimitrios M. |
| author_facet | Jaffke, Lars Morelle, Laure Sau, Ignasi Thilikos, Dimitrios M. |
| contents | We revisit a graph width parameter that we dub bipartite treewidth (btw). Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number, and is closely related to odd-minors. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one "bipartite" vertex, while the width of such decomposition measures the number of "non-bipartite" vertices in a bag. We provide para-NP-completeness results and develop dynamic programming techniques to solve problems on graphs of small btw. In particular, we show that $K_t$-Subgraph-Cover, Weighted Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by btw. We also provide the following dichotomy when $H$ is a 2-connected graph: if $H$ is bipartite, then $H$-Subgraph/Induced-Subgraph/Odd-Minor/Scattered-Packing is para-NP-complete parameterized by btw while, if $H$ is non-bipartite, then the problem is solvable in XP-time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07754 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamic programming on bipartite tree decompositions Jaffke, Lars Morelle, Laure Sau, Ignasi Thilikos, Dimitrios M. Data Structures and Algorithms 05C85, 68R10, 05C75, 05C83, 05C75, 05C69 F.2.2; G.2.2 We revisit a graph width parameter that we dub bipartite treewidth (btw). Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number, and is closely related to odd-minors. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one "bipartite" vertex, while the width of such decomposition measures the number of "non-bipartite" vertices in a bag. We provide para-NP-completeness results and develop dynamic programming techniques to solve problems on graphs of small btw. In particular, we show that $K_t$-Subgraph-Cover, Weighted Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by btw. We also provide the following dichotomy when $H$ is a 2-connected graph: if $H$ is bipartite, then $H$-Subgraph/Induced-Subgraph/Odd-Minor/Scattered-Packing is para-NP-complete parameterized by btw while, if $H$ is non-bipartite, then the problem is solvable in XP-time. |
| title | Dynamic programming on bipartite tree decompositions |
| topic | Data Structures and Algorithms 05C85, 68R10, 05C75, 05C83, 05C75, 05C69 F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2309.07754 |