Confining the Robber on Cographs

TL;DR

This paper investigates the confining cop number in specific graph classes, establishing bounds for planar cographs and $P_5$-free graphs, and explores structural conditions affecting cop strategies.

Contribution

It introduces structural conditions for $P_k$-free graphs that influence confining strategies and determines bounds for planar cographs and $P_5$-free graphs.

Findings

Confined cop number is at most one for planar cographs.

Confined cop number is at most two for planar $P_5$-free graphs.

Connected cographs require at least eight vertices for confining strategies.

Abstract

In this paper, the notions of {\em trapping} and {\em confining} the robber on a graph are introduced. We present some structural necessary conditions for graphs $G$ not containing the path on $k$ vertices (referred to as $P_k$-free graphs) for some $k\ge 4$, so that $k-3$ cops do not have a strategy to capture or confine the robber on $G$. Utilizing such conditions, we show that for planar cographs and planar $P_5$-free graphs the confining cop number is at most one and two, respectively. It is also shown that the number of vertices of a connected cograph on which one cop does not have a strategy to confine the robber has a tight lower-bound of eight. We also explore the effects of twin operations -- which are well known to provide a characterization of cographs -- on the number of cops required to capture or confine the robber on cographs. We conclude by posing two conjectures concerning the confining cop number of $P_5$-free graphs and the smallest planar graph of confining cop number of three.