The radius capture number

TL;DR

The paper studies the radius capture number $\mathrm{rc}(G)$ in the single-cop cop-and-robber game, establishing that $H$ being a retract of $G$ preserves the radius-capture property and deriving tight general bounds $\mathrm{rc}(G) \le \mathrm{rad}(G)-1$ and $\mathrm{rc}(G) \ge \left\lfloor \frac{g(G)}{2} \right\rfloor - 1$. It systematically analyzes $\mathrm{rc}$ across graph families, showing $\mathrm{rc}(G)=\mathrm{rad}(G)-1$ for many vertex-transitive and harmonic-even graphs (including Johnson-type and generalized Johnson graphs, and hypercubes), and provides exact or bound results for outerplanar and Sierpiński graphs. The work also clarifies $\mathrm{rc}$ for various graph products, giving precise formulas for strong, lexicographic, and Cartesian products and establishing exact values for Hamming graphs; it concludes with open questions on characterizing graphs where $\mathrm{rc}(G)=\mathrm{rad}(G)-1$ and links to related invariants. The results advance understanding of how graph structure constrains pursuit-evasion strategies and have potential implications for network security and related combinatorial game theory problems.

Abstract

In the classic cop and robber game, two players--the cop and the robber--take turns moving to a neighboring vertex or staying at their current position. The cop aims to capture the robber, while the robber tries to evade capture. A graph $G$ is called a cop-win graph if the cop can always capture the robber in a finite number of moves. In the cop and robber game with radius of capture $k$, the cop wins if he can come within distance $k$ of the robber. The radius capture number $\rc(G)$ of a graph $G$ is the smallest $k$ for which the cop has a winning strategy in this variant of the game. In this paper, we establish that $\rc(H) \leq \rc(G)$ for any retract $H$ of $G$. We derive sharp upper and lower bounds for the radius capture number in terms of the graph's radius and girth, respectively. Additionally, we investigate the radius capture number in vertex-transitive graphs and identify several families $\cal{F}$ of vertex-transitive graphs with $\rc(G)=\rad(G)-1$ for any $G \in \cal{F}$. We further study the radius capture number in outerplanar graphs, Sierpiński graphs, harmonic even graphs, and graph products. Specifically, we show that for any outerplanar graph $G$, $\rc(G)$ depends on the size of its largest inner face. For harmonic even graphs and Sierpiński graphs $S(n,3)$, we prove that $\rc(G)=\rad(G)-1$. Regarding graph products, we determine exact values of the radius capture number for strong and lexicographic products, showing that they depend on the radius capture numbers of their factors. Lastly, we establish both lower and upper bounds for the radius capture number of the Cartesian product of two graphs.

Figures (2)

• Figure 1: The graph CubicVT[24,6] from the census of small connected cubic vertex-transitive graphs PoSp2013.
• Figure 2: The Sierpiński graph $S(3,3)$.

Theorems & Definitions (38)

• Remark 1
• Theorem 2
• proof
• Theorem 3
• proof
• Corollary 5
• proof
• Theorem 6
• proof
• Theorem 7