The Cop Number of the One-Cop-Moves Game on Planar Graphs
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by
Ziyuan Gao, Boting Yang
2018
Abstract
Cops and robbers is a vertex-pursuit game played on graphs. In the classical
cops-and-robbers game, a set of cops and a robber occupy the vertices of the
graph and move alternately along the graph's edges with perfect information
about each other's positions. If a cop eventually occupies the same vertex as
the robber, then the cops win; the robber wins if she can indefinitely evade
capture. Aigner and Frommer established that in every connected planar graph,
three cops are sufficient to capture a single robber. In this paper, we
consider a recently studied variant of the cops-and-robbers game, alternately
called the one-active-cop game, one-cop-moves game or the lazy-cops-and-robbers
game, where at most one cop can move during any round. We show that Aigner and
Frommer's result does not generalise to this game variant by constructing a
connected planar graph on which a robber can indefinitely evade three cops in
the one-cop-moves game. This answers a question recently raised by Sullivan,
Townsend and Werzanski.
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