Cop and Robber Games When the Robber Can Hide and Ride

  • Authors:

  • Jérémie Chalopin;Victor Chepoi;Nicolas Nisse;Yann Vaxès

  • Affiliations:

  • jeremie.chalopin@lif.univ-mrs.fr and chepoi@lif.univ-mrs.fr and yann.vaxes@lif.univ-mrs.fr;-;nicolas.nisse@inria.fr;-

  • Venue:
  • SIAM Journal on Discrete Mathematics
  • Year:
  • 2011

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Abstract

In the classical cop and robber game, two players, the cop $\mathcal{C}$ and the robber $\mathcal{R}$, move alternatively along edges of a finite graph $G=(V,E)$. The cop captures the robber if both players are on the same vertex at the same moment of time. A graph $G$ is called cop win if the cop always captures the robber after a finite number of steps. Nowakowski and Winkler [Discrete Math., 43 (1983), pp. 235-239] and Quilliot [Problèmes de jeux, de point fixe, de connectivité et de représentation sur des graphes, des ensembles ordonnés et des hypergraphes, Thèse de doctorat d'état, Université de Paris VI, Paris, 1983] characterized the cop-win graphs as graphs admitting a dismantling scheme. In this paper, we characterize in a similar way the class $\mathcal{CWFR}(s,s')$ of cop-win graphs in the game in which the robber and the cop move at different speeds $s$ and $s'$, $s'\leq s$. We also establish some connections between cop-win graphs for this game with $s'1$. In particular, we characterize the graphs which are cop-win for any value of $k$.