The game of pursuit and evasion, when played on graphs, is often referred to as the cops robbers. This classical version has been completely solved by Nowakowski Winkler, who gave exact class graphs for which pursuer can win (cop-win). When graph does not satisfy dismantlability property, Winkler’s Theorem apply. In this paper, we give some weaker notions cop-win robber-win graphs. particular, ...