Guarding isometric subgraphs and Cops and Robber in planar graphs
Combinatorics
2024-06-04 v1 Discrete Mathematics
Abstract
In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.
Keywords
Cite
@article{arxiv.2406.01068,
title = {Guarding isometric subgraphs and Cops and Robber in planar graphs},
author = {Sebastián González Hermosillo de la Maza and Bojan Mohar},
journal= {arXiv preprint arXiv:2406.01068},
year = {2024}
}