The Game of Cops and Robber on (Claw, Even-hole)-free Graphs

TL;DR

This paper proves that in the class of graphs without even holes and claws, two cops are always sufficient to catch the robber, with capture time bounded by twice the number of vertices, advancing understanding of such graph structures.

Contribution

It establishes that the cop number is at most two for claw-free even-hole-free graphs and bounds the capture time, providing new structural insights into these graph classes.

Findings

Cop number is at most two for these graphs.

Capture time is at most 2n rounds.

Results serve as a foundation for further structural study.

Abstract

In this paper, we study the game of cops and robber on the class of graphs with no even hole (induced cycle of even length) and claw (a star with three leaves). The cop number of a graph $G$ is defined as the minimum number of cops needed to capture the robber. Here, we prove that the cop number of all claw-free even-hole-free graphs is at most two and, in addition, the capture time is at most $2n$ rounds, where $n$ is the number of vertices of the graph. Moreover, our results can be viewed as a first step towards studying the structure of claw-free even-hole-free graphs.