Bounding an Optimal Search Path with a Game of Cop and Robber on Graphs (original) (raw)
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We consider the game of Cops and Robber played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to the new parameter called the search-time of the graph. While the search-time is bounded above by the number of vertices, we prove an upper bound of half the number of vertices for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have search-time within a small additive constant of the number of vertices. We consider the ratio of the search-time to the number of vertices, and extend this notion of search-time density to infinite graphs. For the infinite random graph, the search-time density can be any real number in [0, 1]. We also consider the search-time when more than one cop is required to win. We show that for the fixed number of cops, the search-time can be calculated by polynomial algorithm, but it is NP-complete to decide, whether k cops can capture ...
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A key operational problem for those charged with the security of vulnerable facilities (such as airports or art galleries) is the scheduling and deployment of patrols. Motivated by the problem of optimizing randomized, and thus unpredictable, patrols, we present a class of patrolling games on graphs.