1-Guardable Subgraphs of Graphs

The cops and robber game is played on a graph. There are two players in the game, consisting of a set of cops and only one robber. They play, respectively; on each player’s turn, the player may either stay in their vertex or move to an adjacent vertex. The robber is captured when one of the cops enters the vertex with the robber. Therefore, the cop wins the game, and the game ends. In this study, 1-guardable subgraphs of graphs in the game of cops and robber are considered. We mention some special subgraphs and their relations. It is known that if the subgraph is 1-guardable, then it must be isometric, but the converse of this argument may not be true. We show that for the converse to be true, some conditions must be added.