Meyniel's conjecture for the cop number of graphs

Let $G$ be a connected graph with $n$ vertices. The cop number $c(G)$ is the minimum number of cops needed for the Cop Player to have a winning strategy in $G$.

Meyniel's conjecture.

$$c(G)=O(\sqrt{n}).$$

The conjecture is tight up to the constant factor, since there are connected graphs for which $\sqrt{n}/2$ cops are insufficient. Even the weaker assertion that $O(n^{1-\varepsilon})$ cops suffice for some $\varepsilon>0$ remains unknown.