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

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 present results concerning games on G, that is the graph obtained by connecting the corresponding vertices in G and its complement G. In particular we show that for planar graphs c(G) ≤ 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) or c(G − e) > c(G). We show a couple of examples of cop edge-critical graphs having cop number equal to 3.