On edge-periodic graphs, deciding the winner in one-cop-one-robber play takes O(LCM(L) n^3) time, and cycles of length at least 2 l LCM(L) are always robber-win.
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A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity
On edge-periodic graphs, deciding the winner in one-cop-one-robber play takes O(LCM(L) n^3) time, and cycles of length at least 2 l LCM(L) are always robber-win.