On the Cop Number of String Graphs

2024-08-20

International Symposium on Algorithms and Computation

AI Summary

This paper investigates upper bounds on the cop number for string graphs and their subclasses in the Cops and Robber pursuit-evasion game. Addressing the prior bound of 15 for string graphs, we introduce a novel “guarding” strategy that integrates intersection graph structural analysis with geometric graph theory techniques, thereby improving the upper bound to 13. We establish, for the first time, that four cops suffice to capture a robber on any planar graph under the Fully Active variant. Additionally, we tighten the known upper bound on the cop number of two-dimensional box graphs. As a corollary, we prove that string graphs of girth at least five have chromatic number at most 14. The core innovation lies in a combinatorial game-theoretic strategy explicitly tailored to the geometric representation structure of string graphs, significantly advancing extremal bounds for pursuit-evasion games on sparse graph classes.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Search and Optimization: Combinatorial Optimization
• Intelligent Robots: Motion and Path Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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. Gavenv{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 extit{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 $14$.

Problem

• Improve cop number bound for string graphs
• Develop guarding technique for pursuit games
• Address active cops strategy on planar graphs

Innovation

• Developed novel guarding technique
• Improved cop number to 13
• Applied technique to planar graphs