The Localization Game On Cartesian Products

TL;DR

This paper investigates the localization game on Cartesian product graphs, establishing bounds on the localization number and analyzing specific cases like cycles, advancing understanding of pursuit-evasion dynamics in graph products.

Contribution

It provides new bounds for the localization number on Cartesian products and characterizes the number for cycle graphs, contributing to pursuit-evasion game theory.

Findings

stablished ounds for nd nd or artesian nd ycles.

emonstrated ounds are tight in certain cases.

ound or ycles, the localization number is mostly two.

Abstract

The localization game is played by two players: a Cop with a team of $k$ cops, and a Robber. The game is initialised by the Robber choosing a vertex $r \in V$, unknown to the Cop. Thereafter, the game proceeds turn based. At the start of each turn, the Cop probes $k$ vertices and in return receives a distance vector. If the Cop can determine the exact location of $r$ from the vector, the Robber is located and the Cop wins. Otherwise, the Robber is allowed to either stay at $r$, or move to $r'$ in the neighbourhood of $r$. The Cop then again probes $k$ vertices. The game continues in this fashion, where the Cop wins if the Robber can be located in a finite number of turns. The localization number $\zeta(G)$, is defined as the least positive integer $k$ for which the Cop has a winning strategy irrespective of the moves of the Robber. In this paper, we focus on the game played on Cartesian products. We prove that $\zeta( G \square H) \geq \max\{\zeta(G), \zeta(H)\}$ as well as $\zeta(G \square H) \leq \zeta(G) + \psi(H) - 1$ where $\psi(H)$ is a doubly resolving set of $H$. We also show that $\zeta(C_m \square C_n)$ is mostly equal to two.