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. The facility can be thought of as a graph Q of interconnected nodes (e.g. rooms, terminals) and the Attacker can choose to attack any node of Q within a given time T: He requires m consecutive periods there, uninterrupted by the Patroller, to commit his nefarious act (and win). The Patroller can follow any path on the graph. Thus the patrolling game is a win-lose game, where the Value is the probability that the Patroller successfully intercepts an attack, given best play on both sides. We determine analytically optimal (minimax) patrolling strategies for various classes of graphs, and discuss how our results could support decisions about hardening facilities or changing the topology of the terrain to be patrolled
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