Year: 2025
Type: article
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 cop number of a graph is the minimum number of cops required to capture the robber. Gavenčiak 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 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.
DOI: 10.37236/13317
Authors: Harmender Gahlawat, Sandip Das
Institution: Indian Statistical Institute
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Funder: Agence Nationale de la Recherche
OpenAlex ID: W4414785443
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