On Cops and Robbers on $G^{\Xi}$ and cop-edge critical graphs.


  • Domingos Moreira Cardoso Universidade de Aveiro
  • Charles Dominic Universidade de Aveiro
  • Łukasz Witkowski Adam Mickiewicz University
  • Marcin Witkowski Adam Mickiewicz University & University of Oxford




Cops and Robbers, vertex-pursuit games


Cop Robber game is a two player game played on an  undirected graph. In this game cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. In this paper we presents results concerning games on $G^{\Xi}$, that is the graph obtained by connecting the corresponding vertices in $G$ and its complement $\overline{G}$. In particular we show that for planar graphs $c(G^{\Xi})\leq 3$. Furthermore we investigate the cop-edge critical graphs, i.e. graphs that for any edge $e$ in $G$ we have either $c(G-e)<c(G) \text{ or } c(G-e)>c(G)$. We show couple examples of cop-edge critical graphs having cop number equal to~$3$.


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