Subdivisions in the Robber Locating Game

TL;DR

This paper investigates a pursuit game on subdivided graphs, establishing the minimum path length needed for a cop to guarantee catching a robber, extending previous results from complete graphs to general graphs.

Contribution

The authors generalize the known bounds for the cop's winning strategy from complete graphs to all graphs, proving the bound is tight for most cases.

Findings

Cop wins on subdivided graphs if path length m ≥ n/2

Bound is tight for most graph sizes

Extended analysis to graphs with varying edge lengths

Abstract

We consider a game in which a cop searches for a moving robber on a graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any n-vertex graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$ obtained by replacing each edge of $G$ by a path of length $m$, if $m \geqslant n$. They conjectured that this bound was best possible for complete graphs, but the present authors showed that in fact the cop wins on $K^{1/m}$ if and only if $m \geqslant n/2$, for all but a few small values of $n$. In this paper we extend this result to general graphs by proving that the cop has a winning strategy on $G^{1/m}$ provided $m \geqslant n/2$ for all but a few small values of $n$; this bound is best possible. We also consider replacing the edges of $G$ with paths of varying lengths.