On the Cop Number of String Graphs

TL;DR

This paper improves the upper bound on the cop number of string graphs from 15 to 13, introduces a novel guarding technique, and applies it to related graph classes and pursuit-evasion variants.

Contribution

It establishes a tighter bound on the cop number of string graphs and develops a new guarding method applicable to various pursuit-evasion games on representable graphs.

Findings

Cop number of string graphs is at most 13.

Four cops can win the Fully Active Cops and Robber game on planar graphs.

Improved bounds on cop number for boxicity 2 graphs.

Abstract

Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The \emph{cop number} of a graph is the minimum number of cops required to capture the robber. Gaven\v{c}iak et al.~[Eur. J. of Comb. 72, 45--69 (2018)] studied the game on intersection graphs and established that the cop number for the class of string graphs is at most 15, and asked as an open question to improve this bound for string graphs and subclasses of string graphs. We address this question and establish that the cop number of a string graph is at most 13. To this end, we develop a novel \textit{guarding} technique. We further establish that this technique can be useful for other Cops and Robber games on graphs admitting a representation. In particular, we show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al.~[Austr. J. Comb. 76(2), 248--265 (2020)]. In passing, we also improve the known bounds on the cop number of boxicity 2 graphs. Finally, as a corollary of our result on the cop number of string graphs, we establish that the chromatic number of string graphs with girth at least $5$ is at most $14$.