The localization capture time of a graph

TL;DR

This paper introduces the capture time, a new graph parameter measuring the duration of a pursuit-evasion game, and explores its properties, bounds, and conjectures across various graph families.

Contribution

It proposes the capture time as a novel graph parameter, analyzes its properties, and establishes bounds for different graph classes including trees, interval graphs, and incidence graphs.

Findings

Capture time is conjectured to be linear in graph order.

Capture time bounds are established for trees and incidence graphs.

The paper links capture time to treewidth and localization number.

Abstract

The localization game is a pursuit-evasion game analogous to Cops and Robbers, where the robber is invisible and the cops send distance probes in an attempt to identify the location of the robber. We present a novel graph parameter called the capture time, which measures how long the localization game lasts assuming optimal play. We conjecture that the capture time is linear in the order of the graph, and show that the conjecture holds for graph families such as trees and interval graphs. We study bounds on the capture time for trees and its monotone property on induced subgraphs of trees and more general graphs. We give upper bounds for the capture time on the incidence graphs of projective planes. We finish with new bounds on the localization number and capture time using treewidth.