Regular Article: On the Cop Number of a Graph

  • Authors:

  • A. Berarducci;B. Intrigila

  • Affiliations:

  • Univ Pisa, Dipartimento Matemat, Via Buonarroti 2, I 56127 Pisa, Italy and Univ Laquila, Dipartimento Matemat, I 67010 Coppito, Italy;Univ Pisa, Dipartimento Matemat, Via Buonarroti 2, I 56127 Pisa, Italy and Univ Laquila, Dipartimento Matemat, I 67010 Coppito, Italy

  • Venue:
  • Advances in Applied Mathematics
  • Year:
  • 1993

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Abstract

The cop number c(G) of a graph G is an invariant connected with the genus and the girth. We prove that for a fixed k there is a polynomial-time algorithm which decides whether c(G) @? k. This settles a question of T. Andreae. Moreover, we show that every graph is topologically equivalent to a graph with c @? 2. Finally we consider a pursuit-evasion problem in Littlewood's miscellany. We prove that two lions are not always sufficient to catch a man on a plane graph, provided the lions and the man have equal maximum speed. We deal both with a discrete motion (from vertex to vertex) and with a continuous motion. The discrete case is solved by showing that there are plane graphs of cop number 3 such that all the edges can be represented by straight segments of the same length.