Sublinear hitting sets for some geometric graphs

TL;DR

This paper proves the Bollobás-Erdős-Tuza conjecture for various classes of graphs by establishing sublinear hitting sets, especially for geometric and hereditary graphs with large independence numbers, using a unified framework and geometric insights.

Contribution

It introduces a unified framework for finding sublinear hitting sets in locally sparse graphs and extends the conjecture's validity to multiple graph classes, including geometric and hereditary graphs.

Findings

Hitting sets of size O(n / log n) for even-hole-free and disk graphs.

Hitting set size O(√n) for circle graphs with linear independence.

Conjecture holds for hereditary graphs with sublinear separators.

Abstract

For an $n$-vertex graph $G$, let $h(G)$ denote the smallest size of a subset of $V(G)$ such that it intersects every maximum independent set of $G$. A conjecture posed by Bollob\'{a}s, Erd\H{o}s and Tuza in early 90s remains widely open, asserting that for any $n$-vertex graph $G$, if the independence number $\alpha(G) =\Omega(n) $, then $h(G) = o(n)$. In this paper, we establish the validity of this conjecture for various classes of graphs, Our main contributions include: \begin{enumerate} \item We provide a novel unified framework to find sub-linear hitting sets for graphs with certain locally sparse properties. Based on this framework, we can find hitting sets of size at most $O(\frac{n}{\log{n}})$ in any $n$-vertex even-hole-free graph (in particular, chordal graph) and in any $n$-vertex disk graph, with linear independence numbers. \item Utilizing geometric observations and combinatorial arguments, we show that any $n$-vertex circle graph $G$ with linear independence number satisfies $h(G)\le O(\sqrt{n})$. Moreover, we extend this methodology to more general classes of graphs. \item We show the conjecture holds for those hereditary graphs having sublinear balanced separators. \end{enumerate} We also show that $h(G)$ can be upper bounded by constants for several sporadic families of graphs with large independence numbers.