We prove that every connected $P_5$-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected $P_5$-free graph $G$ with independence number at least three contains a three-vertex induced path with vertices $a \hbox{-} b \hbox{-} c$ in order, such that every neighbour of $c$ is also adjacent to one of $a,b$.

翻译：我们证明了每个连通的$P_5$-自由图的警察数至多为二，解决了Sivaraman的一个猜想。为此，我们首先证明每个连通且独立数至少为三的$P_5$-自由图$G$包含一个顶点顺序为$a \hbox{-} b \hbox{-} c$的三顶点诱导路径，使得$c$的每个邻点也与$a,b$之一相邻。