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

TL;DR

This paper extends a key lemma in the cops and robber game to a broader class of graphs, introducing vertebrate graphs as 1-guardable and analyzing their properties.

Contribution

It generalizes Aigner and Fromme's lemma to vertebrate graphs and provides metric characterizations for this larger family, including their cop number.

Findings

Vertebrate graphs are 1-guardable.

Extended lemma applies to a larger class of graphs.

Determined cop number for certain multi-layer generalized Peterson graphs.

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.