Pushing Cops and Robber on Graphs of Maximum Degree 4

TL;DR

This work studies Cops and Robber on oriented graphs with push operations, focusing on the strong-push variant. It shows that transforming the graph via pushes to a DAG yields a cop-win scenario with $c_{sp}(\overrightarrow{G})=1$, and uses this framework to extend known results from subcubic graphs to orientations of 3-degenerate graphs and graphs with $\Delta(G)\le 4$. The main contributions are a constructive push-to-DAG approach and an invariant-based strategy for 4-regular (and thus $\Delta\le 4$) graph orientations, establishing that a single strong-push cop can capture the robber in these broader classes. These results broaden the catalog of graphs for which a single push-enabled cop suffices, and they motivate further questions about chordal digraphs, higher degrees, and potential counterexamples with $c_{sp}>1$.

Abstract

\textsc{Cops and Robber} is a game played on graphs where a set of \textit{cops} aim to \textit{capture} the position of a single \textit{robber}. The main parameter of interest in this game is the \textit{cop 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 \textit{push} operation on a vertex $v$ reverses the orientation of all arcs incident on $v$. We consider a variation of classical \textsc{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 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.

Figures (3)

• Figure 1: An illustration for Case 1 of \ref{['C:edge']}. Once $\mathcal{R}$ moves to $x$, $\mathcal{C}$ traps $\mathcal{R}$ using \ref{['L:trivial']}
• Figure 2: An illustration for Case 2 of \ref{['C:edge']}. We illustrate the case when $w\in N^+(x) \cap N^+(y)$. \ref{['fig:OC35']} illustrates two steps: $\mathcal{R}$ moving from $y$ to $x$ and then $\mathcal{C}$ pushing $w$. Once $\mathcal{R}$ moves to $u$ from $x$, $\mathcal{C}$ traps $\mathcal{R}$ using \ref{['L:trivial']}
• Figure 5: A chordal digraph can be pushed to be a DAG iff it does not contain any of the two orientations of $K_4$ as subdigraphs.

Theorems & Definitions (24)

• Theorem 1.1
• Proposition 1.2: das2023cops
• Corollary 1.3
• proof
• Lemma 1.3
• Theorem 1.4
• Proposition 2.1: das2023cops
• Lemma 3.1
• proof
• Lemma 3.1