Abstract
Let G be a finite connected graph of order at least two. Two players (a cop and a robber) are allowed to play a game on G according to the following rule: The cop chooses a vertex to stay then the robber chooses a vertex afterwards. After that they move alternately along edges of G, where the robber may opt to stay put when his turn to move comes. The cop wins if he succeeds in putting himself on top of the robber, otherwise, the robber wins. A graph G is said to be a cop-win graph if the cop has a winning strategy on it. Otherwise, G is called a robber-win graph. In this paper we give necessary and sufficient conditions for the join, corona, and lexicographic product of two connected graphs to be cop-win graphs. It is shown that the cartesian product G × H of any connected graphs G and H of orders at least two is a robber-win graph.
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