Layered tree-independence number and clique-based separators

TL;DR

This paper explores the relationship between layered tree-independence number and clique-based separators in geometric graphs, providing bounds and algorithmic implications for weighted problems.

Contribution

It establishes bounds on layered tree-independence number for various geometric intersection graphs and derives subexponential algorithms for key weighted problems.

Findings

Layered tree-independence number is $ ext{O}(g)$ for $g$-map graphs.

It is $ ext{O}(rac{r}{ anh r})$ for hyperbolic uniform disk graphs.

It is $ ext{O}(1)$ for spherical uniform disk graphs.

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.