A Vertex-To-Vertex Pursuit Game Played with Disjoint Sets of Edges

The problem is to determine the number of ‘cops’ needed to capture a ‘robber’ where the game is played with perfect information, the different sides moving alternately. The ‘cops’ capture the ‘robber’ if one of them occupies the same vertex as the robber at any time in the game. Normally, both sides can move along any edge present in the graph. We investigate the game where the two sides move along disjoint sets of edges. Two natural situations occur. One, given a graph, the cops move along the edges of the graph and the robber moves along the complementary edges. Two, the adversaries can move along disjoint sets of edges present in a product of graphs.

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