Sublinear hitting sets for some geometric graphs

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.