Layered tree-independence number and clique-based separators
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| Format: | Preprint |
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2025
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| author | Dallard, Clément Milanič, Martin Munaro, Andrea Yang, Shizhou |
| author_facet | Dallard, Clément Milanič, Martin Munaro, Andrea Yang, Shizhou |
| contents | Motivated by a question of Galby, Munaro, and Yang (SoCG 2023) asking whether every graph class of bounded layered tree-independence number admits clique-based separators of sublinear weight, we investigate relations between layered tree-independence number, weight of clique-based separators, clique cover degeneracy and independence degeneracy. In particular, we provide a number of results bounding these parameters on geometric intersection graphs. For example, we show that the layered tree-independence number is $\mathcal{O}(g)$ for $g$-map graphs, $\mathcal{O}(\frac{r}{\tanh r})$ for hyperbolic uniform disk graphs with radius $r$, and $\mathcal{O}(1)$ for spherical uniform disk graphs with radius $r$. Our structural results have algorithmic consequences. In particular, we obtain a number of subexponential or quasi-polynomial-time algorithms for weighted problems such as \textsc{Max Weight Independent Set} and \textsc{Min Weight Feedback Vertex Set} on several geometric intersection graphs. Finally, we conjecture that every fractionally tree-independence-number-fragile graph class has bounded independence degeneracy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Layered tree-independence number and clique-based separators Dallard, Clément Milanič, Martin Munaro, Andrea Yang, Shizhou Combinatorics Computational Geometry Discrete Mathematics 05C10, 05C62, 05C75, 05C85 Motivated by a question of Galby, Munaro, and Yang (SoCG 2023) asking whether every graph class of bounded layered tree-independence number admits clique-based separators of sublinear weight, we investigate relations between layered tree-independence number, weight of clique-based separators, clique cover degeneracy and independence degeneracy. In particular, we provide a number of results bounding these parameters on geometric intersection graphs. For example, we show that the layered tree-independence number is $\mathcal{O}(g)$ for $g$-map graphs, $\mathcal{O}(\frac{r}{\tanh r})$ for hyperbolic uniform disk graphs with radius $r$, and $\mathcal{O}(1)$ for spherical uniform disk graphs with radius $r$. Our structural results have algorithmic consequences. In particular, we obtain a number of subexponential or quasi-polynomial-time algorithms for weighted problems such as \textsc{Max Weight Independent Set} and \textsc{Min Weight Feedback Vertex Set} on several geometric intersection graphs. Finally, we conjecture that every fractionally tree-independence-number-fragile graph class has bounded independence degeneracy. |
| title | Layered tree-independence number and clique-based separators |
| topic | Combinatorics Computational Geometry Discrete Mathematics 05C10, 05C62, 05C75, 05C85 |
| url | https://arxiv.org/abs/2506.12424 |