Cop number of $2K_2$-free graphs

TL;DR

This paper investigates the cop number in $2K_2$-free graphs, establishing upper bounds under certain conditions and conjecturing a universal bound of 2 for all such graphs.

Contribution

It proves upper bounds on the cop number for specific classes of $2K_2$-free graphs and proposes a conjecture for the general case.

Findings

Cop number is at most 2 for graphs with diameter 3.

Cop number is at most 2 if the graph lacks certain induced cycles.

Conjecture that all $2K_2$-free graphs have cop number at most 2.

Abstract

We prove that the cop number of a $2K_2$-free graph is at most $2$ if it has diameter $3$ or does not have an induced cycle of length $k$, where $k \ \in \{3,4,5\}$. We conjecture that the cop number of every $2K_2$-free graph is at most $2$.