Primal-Dual Cops and Robber

TL;DR

This work introduces Primal-Dual Cops and Robber, a planar-graph pursuit game where cops occupy faces and the robber occupies vertices, with capture when all faces incident to the robber’s vertex are occupied. The authors prove a sharp Δ-based dichotomy: the primal-dual cop number is bounded for planar graphs with maximum degree Δ up to 4 (specifically $c^*(G)\le 3$ for $\Delta=3$ and $c^*(G)\le 6$ for $\Delta=4$, both tight), while for Δ=5 there exist infinitely many graphs with $c^*(G)=\Omega(\sqrt{\log n})$, showing unbounded growth. The Δ=3 result relies on a three-face-cop strategy, the Δ=4 case uses a dual-graph strategy to bound from above by 6 (and provides a tight lower bound via a dodecahedron-based construction), and the Δ=5 lower bound leverages a grid-like construction combined with a fast-robber-type strategy to simulate known evasive results. Overall, the paper delineates a complete Δ-dependent landscape for planar graphs and highlights intriguing directions for higher degrees and non-planar extensions, including potential connections to cycle-double-cover concepts. The results have implications for pursuit-evasion in planar networks and provide precise thresholds distinguishing bounded and unbounded regimes.

Abstract

Cops and Robber is a family of two-player games played on graphs in which one player controls a number of cops and the other player controls a robber. In alternating turns, each player moves (all) their figures. The cops try to capture the robber while the latter tries to flee indefinitely. In this paper we consider a variant of the game played on a planar graph where the robber moves between adjacent vertices while the cops move between adjacent faces. The cops capture the robber if they occupy all incident faces. We prove that a constant number of cops suffices to capture the robber on any planar graph of maximum degree $Δ$ if and only if $Δ\leq 4$.

Figures (4)

• Figure 2: A vertex-cop and its four accompanying face-cops moving from $u$ to $v$.
• Figure 3: The $5$-times subdivided dodecahedron graph $D_5$ (left) and an illustration of the strategy in \ref{['lem:5_subdivided_dodecahedron']} (right) with two exemplary positions for the cop $c_2$ at distance $9$ to $w_1$.
• Figure 4: Left: A portion of the $4$-regular planar graph $G$ based on $D_5$. For the sake of readability only some $30$-cycles are shown in the faces of $D$. The subgraph $G_0$ is highlighted in thick. Top-right: The degree-based split of $v$. Bottom-right: Adjoining a $30$-cycle to a $30$-cycle.
• Figure 5: $G_{4,2,2}$: A $4 \times 4$ grid with each edge subdivided four times and two rings. Faces are colored according to their closest grid vertex. Deep and shallow faces are light and dark, respectively.

Theorems & Definitions (7)

• Theorem 1
• Remark 2: Connectedness
• Remark 4
• Lemma 5
• Theorem 6: Fomin2010_FastRobber
• Lemma 7
• Lemma 8