Abstract
AbstractAigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction that shows that a sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar‐directed graph.
Highlights
The general study of pursuit games on graphs drew a substantial amount of research attention over the last decade
Dynamic processes are typically already significantly more difficult to analyze than properties of static graphs, and game theoretic interactions between opposing parties drive the complexity to another level
The main question is to determine, for each graph, the minimum value of k for which there is a strategy for the cops that guarantees a win within finite time
Summary
The general study of pursuit games on graphs drew a substantial amount of research attention over the last decade. The main question is to determine, for each graph, the minimum value of k (known as the cop number of the graph) for which there is a strategy for the cops that guarantees a win within finite time This game-theoretic graph invariant was introduced by Aigner and Fromme [1] shortly after the game’s appearance in the combinatorial literature, and in that same paper, the authors proved the elegant and sharp result that every planar graph has cop number at most three. We close with a short proof of Proposition 1.1 in the final section
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