The Cop Number of Graphs with Forbidden Induced Subgraphs
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In the game of Cops and Robber, a team of cops attempts to capture a robber on a graph G. Initially, all cops occupy some vertices in G and the robber occupies another vertex. In each round, a cop can move to one of its neighbors or stay idle, after which the robber does the same. The robber is caught by a cop if the cop lands on the same vertex which is currently occupied by the robber. The minimum number of cops needed to guarantee capture of a robber on G is called the cop number of G, denoted by c(G). We say a family F of graphs is cop-bounded if there is a constant M so that c(G)≤ M for every graph G∈ F. Joret, Kaminński, and Theis [Contrib. Discrete Math. 2010] proved that the class of all graphs not containing a graph H as an induced subgraph is cop-bounded if and only if H is a linear forest; morerover, C(G)≤ k-2 if if G is induced-P_k-free for k≥ 3. In this paper, we consider the cop number of a family of graphs forbidding certain two graphs and generalized some previous results.
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