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While finite cop-win finite graphs possess a good structural characterization, none is known for infinite cop-win graphs. As evidence that such a characterization might not exist, we provide as large as possible classes of infinite graphs with finite cop number. More precisely, for each infinite cardinal and each positive integer k, we construct 2 non-isomorphic k-cop-win graphs satisfying additional properties such Contract grant sponsors: NSERC; MITACS. Journal of Graph Theory ᭧doi:10.1002/jgt.20484 fatcat:lyttmtzlxfdtti7bp6kvr3sleu