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arXiv · 1804.02802

A note on 1-guardable graphs in the cops and robber game

Linyuan Lu·Zhiyu Wang
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Abstract

In the cops and robber games played on a simple graph $G$, Aigner and Fromme's lemma states that one cop can guard a shortest path in the sense that the robber cannot enter this path without getting caught after finitely many steps. In this paper, we extend Aigner and Fromme's lemma to cover a larger family of graphs and give metric characterizations of these graphs. In particular, we show that a generalization of block graphs, namely vertebrate graphs, are 1-guardable. We use this result to give the cop number of some special class of multi-layer generalized Peterson graphs.

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Linyuan Lu, Zhiyu Wang. 2018-04-10. A note on 1-guardable graphs in the cops and robber game. https://arxiv.org/abs/1804.02802

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