A game of cops and robbers.

*(English)*Zbl 0539.05052Let G be a finite connected undirected graph. Two players, the cop C and robber R, play a game on G according to the following rules. First C then R occupy some vertex of G. After that they move alternately along edges of G. The cop wins if he succeeds in eventually occupying the same vertex as R, otherwise R wins. A graph G is ”cop-win” if C has a winning strategy; otherwise it is ”robber-win”. R. Novakowski and P. Winkler [Discrete Math. 43, 235-239 (1983; Zbl 0508.05058)] previously gave an algorithmic characterization of cop-win graphs, and showed that this family formed a ”variety” in the sense of closure under (strong) graph products and retractions. The Novakowski-Winkler results are first reviewed, and then the authors generalize the game to a game where a team of n cops chase the robber. The ”cop number” c(G) is defined to be the minimum number of cops required to catch the robber in G. Graphs \(G_ n\) are constructed for which \(c(G_ n)\geq n\), namely n-regular graphs with girth at least 5. The most striking result proved is that c(G)\(\leq 3\) for all planar graphs G.

Reviewer: T.D.Parsons

##### MSC:

05C57 | Games on graphs (graph-theoretic aspects) |

05C75 | Structural characterization of families of graphs |

91A24 | Positional games (pursuit and evasion, etc.) |

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\textit{M. Aigner} and \textit{M. Fromme}, Discrete Appl. Math. 8, 1--11 (1984; Zbl 0539.05052)

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