On 1-Planar Graphs with Bounded Cop-Number

TL;DR

This work determines that the cop-number of 1-planar graphs is bounded under restricted crossing patterns: if a 1-planar graph can be embedded without any $ imes$-crossings, then $c(G) \\le 21$, and if at most $ imes$-crossings occur, then $c(G) \\le \gamma+21$. The authors achieve this via guarding strategies that extend planar techniques, introducing two configurations, $\mathbb{P}$-Configuration (path guarding) and $\mathbb{C}$-Configuration (cycle guarding), implemented by three seven-cop sets and a five-cop path-guarding mechanism derived from a shortest path. A central tool is guarding crossing points and subgraphs in the planarised drawing $G^\times$, using kite edges to preserve guardability and maintain a well-defined robber territory. The results generalize the classical planar bound (three cops) to a broader class of 1-planar graphs, and they also bound the cop-number in terms of the number of $ imes$-crossings, offering a concrete pathway toward classifying 1-planar graphs by cop-number. The work thus advances understanding of pursuit-evasion on beyond-planar graphs and highlights crossing-structure as a critical obstacle in extending planar strategies.

Abstract

Cops and Robbers is a type of pursuit-evasion game played on a graph where a set of cops try to capture a single robber. The cops first choose their initial vertex positions, and later the robber chooses a vertex. The cops and robbers make their moves in alternate turns: in the cops' turn, every cop can either choose to move to an adjacent vertex or stay on the same vertex, and likewise the robber in his turn. If the cops can capture the robber in a finite number of rounds, the cops win, otherwise the robber wins. The cop-number of a graph is the minimum number of cops required to catch a robber in the graph. It has long been known that graphs embedded on surfaces (such as planar graphs and toroidal graphs) have a small cop-number. Recently, Durocher et al. [Graph Drawing, 2023] investigated the problem of cop-number for the class of $1$-planar graphs, which are graphs that can be embedded in the plane such that each edge is crossed at most once. They showed that unlike planar graphs which require just three cops, 1-planar graphs have an unbounded cop-number. On the positive side, they showed that maximal 1-planar graphs require only three cops by crucially using the fact that the endpoints of every crossing in an embedded maximal 1-planar graph induce a $K_4$. In this paper, we show that the cop-number remains bounded even under the relaxed condition that the endpoints induce at least three edges. More precisely, let an $\times$-crossing of an embedded 1-planar graph be a crossing whose endpoints induce a matching; i.e., there is no edge connecting the endpoints apart from the crossing edges themselves. We show that any 1-planar graph that can be embedded without $\times$-crossings has cop-number at most 21. Moreover, any 1-planar graph that can be embedded with at most $γ$ $\times$-crossings has cop-number at most $γ+ 21$.

Figures (9)

• Figure 1: An example that illustrates the insertion of kite edges in a graph $G \in \hat{\mathcal{G}}$. The edges $(a,c)$ and $(c,d)$ are duplicated and inserted as uncrossed kite edges at the crossings $\{(a,b), (c,d)\}$ and $\{(a,c), (e,d)\}$ respectively. The kite edge $(f,g)$ is common to both the crossings $\{(e,g), (a,f)\}$ and $\{(g,h), (f,c)\}$.
• Figure 2: An illustration of cop and robber territory for the graph from Figure \ref{['fig: adding kite edges']}. Let $H$ be the path $(a,c,z,b)$ of $G^\times$ that is guarded. The cop territory $\mathcal{C}(H)$ consists of edges $\{(a,b), (c,d), (f,a), (f,c), e_1, e_2\}$. The robber is on the vertex $f$ and the robber territory $\mathcal{R}(H)$ consists of the edges $\{(g,e), (g,f), (g,h), (e,d)\}$.
• Figure 3: An example to illustrate 'adjacency' to the robber territory. Here, $H$ is a path, and $S_H(u) = \{(u,a), (u,b)\}$ (even though $(u,c) \in E(G)$), $S_H(v) = \{(v,c)\}$ and $S_H(w) = \{(w,d), (w,e)\}$.
• Figure 4: An illustration for Lemma \ref{['lem: five cops guard a path']}. The solid edges form a shortest $(a,b)$-path $P$. The dashed edges belong to $G$, but not to $P$. The dotted region shows vertices of $P$ that belong to $Q$.
• Figure 5: An illustration of a cycle $D = [P_1, P_2]$ guarded in $\mathbb{C}$-Configuration. The dotted region shows the robber territory $\mathcal{R}(D)$.

Theorems & Definitions (32)

• Definition 2: Cop Territory
• Definition 3: Robber Territory
• proof
• proof
• Lemma 6
• proof
• Corollary 8
• Corollary 9
• proof
• Corollary 10