Pushing Cops and Robber on graphs of maximum degree four

Cops and Robber is a game played on graphs where a set of cops aim to capture the position of a single robber . The main parameter of interest in this game is the cop number , which is the minimum number of cops that are sufficient to guarantee the capture of the robber. In a directed graph G ⃗ , the push operation on a vertex v reverses the orientation of all arcs incident to v . We consider a variation of the classical Cops and Robber 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 out-neighbor in its turn. [Das et al., CALDAM, 2023] introduced this variant and established that if 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 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. Moreover, we establish that if G ⃗ can be made to be a directed acyclic graph, then one cop with push ability has a winning strategy.