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

TL;DR

This paper establishes a new bound on the cop number in the cops and robber game based on the 2-component order connectivity of the graph, linking graph structure to pursuit-evasion complexity.

Contribution

It introduces a novel bound on the cop number using the 2-component order connectivity, connecting graph connectivity measures to pursuit strategies.

Findings

Provides a bound on cop number in terms of 2-component order connectivity

Links graph structural properties to pursuit-evasion game complexity

Offers insights into graph parameters influencing cop and robber dynamics

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.