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Cop-Win Graphs: Optimal Strategies and Corner Rank
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<span title="2017-03-10">2017</span>
<i >
arXiv
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<span class="release-stage" >pre-print</span>
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<span title="2016-07-12">2016</span>
<span class="release-stage" >pre-print</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1607.03471v1">arXiv:1607.03471v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nflotzje2vdj3kikvh7wy4ltfq">fatcat:nflotzje2vdj3kikvh7wy4ltfq</a> </span>
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We investigate the game of cops and robber, played on a finite graph, between one cop and one robber. If the cop can force a win on a graph, the graph is called cop-win. We describe a procedure we call corner ranking, performed on a graph, which assigns a positive integer or ∞ to each vertex. We give a characterization of cop-win in terms of corner rank and also show that the well-known characterization of cop-win via dismantling orderings follows from our work. From the corner rank we can
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... mine the capture time of a graph, i.e. the number of turns the cop needs to win. We describe a class of optimal cop strategies we call Lower Way strategies, and a class of optimal robber strategies we call Higher Way strategies. Roughly speaking, in a Lower Way strategy, the cop pushes the robber down to lower ranked vertices, while in a Higher Way strategy, the robber moves to a highest rank vertex that is "safe." While interesting in their own right, the strategies are themselves tools in our proofs. We investigate various properties of the Lower Way strategies.
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