Localization game capture time of trees and outerplanar graphs
Vesna Iršič Chenoweth, Matija Skrt
TL;DR
The paper analyzes the localization game and the Localization Capture Time Conjecture, delivering improved upper bounds for the trees and proving LCTC for a subclass of outerplanar graphs with $\zeta(G)=2$. It introduces a coloring-based generalization equivalent to the original game and shows that its height equals $\mathrm{lcapt}_k(G) + 1$, offering a new path to universal bounds such as $\mathrm{lcapt}(G) \le n$. By connecting exact results on trees and outerplanar graphs to this broader framework, the work lays groundwork for scaling bounds toward the conjectured linear behavior in graph order $n$. Overall, the results sharpen understanding of capture times on specific graph families and propose a versatile method for approaching LCTC in broader classes.
Abstract
The localization game is a variant of the game of Cops and Robber in which the robber is invisible and moves between adjacent vertices, but the cops can probe any $k$ vertices of the graph to obtain the distance between probed vertices and the robber. The localization number of a graph is the minimum $k$ needed for cops to be able to locate the robber in finite time. The localization capture time is the number of rounds needed for cops to win. The localization capture time conjecture claims that there exists a constant $C$ such that the localization number of every connected graph on $n$ vertices is at most $Cn$. While it is known that the conjecture holds for trees, in this paper we significantly improve the known upper bound for the localization capture time of trees. We also prove the conjecture for a subclass of outerplanar graphs and present a generalization of the localization game that appears useful for making further progress towards the conjecture.