Sparse Outerstring Graphs Have Logarithmic Treewidth
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913404511322112 |
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| author | An, Shinwoo Oh, Eunjin Xue, Jie |
| author_facet | An, Shinwoo Oh, Eunjin Xue, Jie |
| contents | An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with $n$ vertices has treewidth $O(α\log n)$, where $α$ denotes the arboricity of the graph, with an almost matching lower bound of $Ω(α\log (n/α))$. As a corollary, we show that a $t$-biclique-free outerstring graph has treewidth $O(t(\log t)\log n)$. This leads to polynomial-time algorithms for most of the central NP-complete problems such as \textsc{Independent Set}, \textsc{Vertex Cover}, \textsc{Dominating Set}, \textsc{Feedback Vertex Set}, \textsc{Coloring} for sparse outerstring graphs. Also, we can obtain subexponential-time (exact, parameterized, and approximation) algorithms for various NP-complete problems such as \textsc{Vertex Cover}, \textsc{Feedback Vertex Set} and \textsc{Cycle Packing} for (not necessarily sparse) outerstring graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17424 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sparse Outerstring Graphs Have Logarithmic Treewidth An, Shinwoo Oh, Eunjin Xue, Jie Computational Geometry Data Structures and Algorithms An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with $n$ vertices has treewidth $O(α\log n)$, where $α$ denotes the arboricity of the graph, with an almost matching lower bound of $Ω(α\log (n/α))$. As a corollary, we show that a $t$-biclique-free outerstring graph has treewidth $O(t(\log t)\log n)$. This leads to polynomial-time algorithms for most of the central NP-complete problems such as \textsc{Independent Set}, \textsc{Vertex Cover}, \textsc{Dominating Set}, \textsc{Feedback Vertex Set}, \textsc{Coloring} for sparse outerstring graphs. Also, we can obtain subexponential-time (exact, parameterized, and approximation) algorithms for various NP-complete problems such as \textsc{Vertex Cover}, \textsc{Feedback Vertex Set} and \textsc{Cycle Packing} for (not necessarily sparse) outerstring graphs. |
| title | Sparse Outerstring Graphs Have Logarithmic Treewidth |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2406.17424 |