Cops and Robbers on \(P_5\)-Free Graphs

.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 \(a - b - c\) in order, such that every neighbor of \(c\) is also adjacent to one of \(a,b\).Keywordscops and robberscop numberforbidden induced subgraph\(P_5\)-free graphMSC codes05C57