A bound for the cops and robber problem in terms of 2-component order connectivity

TL;DR

This work studies the cops and robber game on graphs, introducing the $2$-component order connectivity ($2$-coc$(G)$) as a structural parameter. The authors develop a reduction-based approach with rules RR1–RR3 that guard vertices and shrink the graph to $G'$ where components outside the removed set have size at most $2$, confining the robber to a connected component $H$. They show $c(G) \le r + \max_H c(H)$ and, within each $H$, derive $c(H) \le \frac{|U'|}{3} + 4$ by a three-cop strategy together with a vertex-cover bound on a derived subgraph, yielding the bound $c(G) \le \frac{|U|}{3} + 4$. This establishes a concrete link between the cop number and the $2$-coc parameter, contributing a Meyniel-type bound in terms of a structural graph parameter and suggesting directions for tighter coc-based bounds. The results offer a framework for parameterized analysis of pursuit-evasion on graphs and potential improvements to constants in related bounds.

Abstract

In the cops and robber game, there are multiple cops and a single robber taking turns moving along the edges of a graph. The goal of the cops is to capture the robber (move to the same vertex as the robber) and the goal of the robber is to avoid capture. The cop number of a given graph is the smallest number of cops required to ensure the capture of the robber. The k-component order connectivity of a graph G = (V, E) is the size of a smallest set U, such that all the connected components of the induced graph on V \ U are of size at most k. In this brief note, we provide a bound on the cop number of graphs in terms of their 2-component order connectivity.

Figures (1)

• Figure 1: Proof of Lemma \ref{['ll:l2']}. The vertex $r$ is the first instance when $\mathcal{R}$ moves outside of $U'_H$. Assuming that $r$ is in a component of size 2 in $G'[V \setminus U]$, we let $y$ be the neighbor of $r$ outside $U'_H$. The vertices $a, b$ are the two neighbors of $r$ in $U'_H$. The vertices $d, e$ are the two neighbors of $y$ in $U'_H$.

Theorems & Definitions (5)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof