  \textsc{Cops and Robber} is one of the most studied two-player
pursuit-evasion games played on graphs, where multiple \textit{cops},
controlled by one player, pursue a single \textit{robber}. The main parameter
of interest is the \textit{cop number} of a graph, which is the minimum number
of cops that can ensure the \textit{capture} of the robber.
  \textsc{Cops and Robber} is also well-studied on directed/oriented graphs. In
directed graphs, two kinds of moves are defined for players: \textit{strong
move}, where a player can move both along and against the orientation of an arc
to an adjacent vertex; and \textit{weak move}, where a player can only move
along the orientation of an arc to an \textit{out-neighbor}. We study three
variants of \textsc{Cops and Robber} on oriented graphs: \textit{strong cop
model}, where the cops can make strong moves while the robber can only make
weak moves; \textit{normal cop model}, where both cops and the robber can only
make weak moves; and \textit{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.
