Cop-win graphs with maximum capture-time

We present an upper bound n−4n−4 for the maximum length of a cop and robber game (the capture-time  ) on a cop-win graph of order nn. This bound matches the known lower bound. We analyze the structure of the class of all graphs attaining this maximum and describe an inductive construction of the entire class.A cop and robber game is a two-player vertex-to-vertex pursuit combinatorial game where the players stand on the vertices of a graph and alternate in moving to adjacent vertices. Cop’s goal is to capture the robber by occupying the same vertex as the robber, robber’s goal is to avoid capture.