Pushing Cops and Robber on Graphs of Maximum Degree 4

$\text{textsc}CopsandRobber$ is a game played on graphs where a set of $\mathit{\text{cops}}$ aim to $\mathit{\text{capture}}$ the position of a single $\mathit{\text{robber}}$. The main parameter of interest in this game is the $\mathit{\text{cop}}\mathit{\text{number}}$, which is the minimum number of cops that are sufficient to guarantee the capture of the robber. In a directed graph $\overrightarrow{G}$, the $\mathit{\text{push}}$ operation on a vertex $v$ reverses the orientation of all arcs incident on $v$. We consider a variation of classical $\text{textsc}CopsandRobber$ on oriented graphs, where in its turn, each cop can either move to an out-neighbor of its current vertex or push some vertex of the graph, whereas, the robber can move to an adjacent vertex in its turn. [Das et al., CALDAM, 2023] introduced this variant and established that if $\overrightarrow{G}$ is an orientation of a subcubic graph, then one cop with push ability has a winning strategy. We extend these results to establish that if $\overrightarrow{G}$ is an orientation of a $3$-degenerate graph, or of a graph with maximum degree $4$, then one cop with push ability has a winning strategy.