Confining the Robber on Cographs

In this paper the notions of {\em trapping} and {\em confining} the robber on a graph are introduced. We present some necessary conditions for graphs $G$ not containing the path on $k$ vertices (for some $k\ge 4$) as an induced subgraph so that $k-3$ cops do not have a winning strategy on $G$. Utilizing the latter conditions together with a characterization of cographs, we show that on cograph the robber can always be confined by one cop. We conclude by posing a conjecture about the confining cop number of $P_k$-free graphs.