Cops and Robbers, Clique Covers, and Induced Cycles

TL;DR

This paper explores the relationships between the cop number, independence number, and clique cover number in graphs, proving the existence of graphs where these parameters are equal and analyzing structural properties of graphs with equal cop and clique cover numbers.

Contribution

It demonstrates, using random graphs, the existence of graphs where the cop number, independence number, and clique cover number are all equal, and characterizes the structure of graphs with equal cop and clique cover numbers.

Findings

Existence of graphs with c(G)=α(G)=θ(G)=k for all k≥1

Graphs with c(G)=θ(G)≥3 contain induced cycles of all lengths 3 to k+1

Perfect graphs with α(G)≥4 have c(G)<α(G)

Abstract

We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$, the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(G)$. For all graphs $G$, $c(G) \leq \alpha(G) \leq \theta(G)$ where $\alpha(G)$ and $\theta(G)$ are the independence number and clique cover number respectively. In 2022 Turcotte asked if $c(G) < \alpha(G)$ for all graphs with $\alpha(G) \geq 3$. Recently, Char, Maniya, and Pradhan proved this is false, at least when $\alpha = 3$,by demonstrating the compliment of the Shrikhande graph has cop number and independence number $3$. We prove, using random graphs, the stronger result that for all $k\geq 1$ there exists a graph $G$ such that $c(G) = \alpha(G) = \theta(G) = k$. Next, we consider the structure of graphs with $c(G) = \theta(G) \geq 3$. We prove, using structural arguments, that any graphs $G$ which satisfies $c(G) = \theta(G) = k \geq 3$ contain induced cycles of all lengths $3\leq t \leq k+1$. This implies all perfect graphs $G$ with $\alpha(G)\geq 4$ have $c(G) < \alpha(G)$. Additionally,we discuss if typical triangle-free and $C_4$-free graphs will have $c(G) < \alpha(G)$.