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arXiv · 2506.12424

Layered tree-independence number and clique-based separators

Clément Dallard·Martin Milanič·Andrea Munaro·Shizhou Yang
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Abstract

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.

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Clément Dallard, Martin Milanič, Andrea Munaro, Shizhou Yang. 2025-06-14. Layered tree-independence number and clique-based separators. https://arxiv.org/abs/2506.12424

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