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\textsc{Cops and Robber} is one of the most studied two-playerpursuit-evasion games played on graphs, where multiple \textit{cops},controlled by one player, pursue a single \textit{robber}. The main parameterof interest is the \textit{cop number} of a graph, which is the minimum numberof cops that can ensure the \textit{capture} of the robber.  \textsc{Cops and Robber} is also well-studied on directed/oriented graphs. Indirected graphs, two kinds of moves are defined for players: \textit{strongmove}, where a player can move both along and against the orientation of an arcto an adjacent vertex; and \textit{weak move}, where a player can only movealong the orientation of an arc to an \textit{out-neighbor}. We study threevariants of \textsc{Cops and Robber} on oriented graphs: \textit{strong copmodel}, where the cops can make strong moves while the robber can only makeweak moves; \textit{normal cop model}, where both cops and the robber can onlymake weak moves; and \textit{weak cop model}, where the cops can make weakmoves while the robber can make strong moves. We study the cop number of thesemodels with respect to several variants of retracts on oriented graphs andestablish that the strong and normal cop number of an oriented graph remainsinvariant in their strong and distributed retracts, respectively. Next, we goon to study all three variants with respect to the subdivisions of graphs andoriented graphs. Finally, we establish that all these variants remaincomputationally difficult even when restricted to the class of 2-degeneratebipartite graphs.

Cops and robber on variants of retracts and subdivisions of oriented graphs