Cops and Robbers, Clique Covers, and Induced Cycles

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 $c(G) < \alpha(G)$.