Localization game capture time of trees and outerplanar graphs

TL;DR

This paper investigates the localization game on trees and outerplanar graphs, improving bounds on capture time and supporting the conjecture that localization number scales linearly with graph size.

Contribution

It significantly improves the upper bound for capture time on trees, proves the conjecture for a subclass of outerplanar graphs, and introduces a generalized game framework.

Findings

Improved upper bound for localization capture time on trees

Proved the localization conjecture for a subclass of outerplanar graphs

Presented a generalized localization game for future research

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.