Cops and robber on variants of retracts and subdivisions of oriented graphs
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Cops and Robber is one of the most studied two-player pursuit-evasion games played on graphs, where multiple cops, controlled by one player, pursue a single robber. The main parameter of interest is the cop number of a graph, which is the minimum number of cops that can ensure the capture of the robber. Cops and Robber is also well-studied on directed/oriented graphs. In directed graphs, two kinds of moves are defined for players: strong move, where a player can move both along and against the orientation of an arc to an adjacent vertex; and weak move, where a player can only move along the orientation of an arc to an out-neighbor. We study three variants of Cops and Robber on oriented graphs: strong cop model, where the cops can make strong moves while the robber can only make weak moves; normal cop model, where both cops and the robber can only make weak moves; and weak cop model, where the cops can make weak moves while the robber can make strong moves. We study the cop number of these models with respect to several variants of retracts on oriented graphs and establish that the strong and normal cop number of an oriented graph remains invariant in their strong and distributed retracts, respectively. Next, we go on to study all three variants with respect to the subdivisions of graphs and oriented graphs. Finally, we establish that all these variants remain computationally difficult even when restricted to the class of 2-degenerate bipartite graphs.
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