Locating Robber with Cop Strategy Graph: Subdivision vs. Multiple Cop

TL;DR

This paper introduces new constructions for the Robber Locating Game, analyzing the relationship between localization and subdivision numbers, and clarifies their non-linear connection through theoretical proofs.

Contribution

It proposes state variable and strategy graph constructions to analyze the game, and clarifies the non-linear relationship between localization and subdivision numbers.

Findings

Established a clear non-linear relationship between localization and subdivision numbers.

Revisited and consolidated previous proofs on graph parameters.

Provided explicit results and algorithms for the game analysis.

Abstract

We consider the Robber Locating Game, where an invisible moving robber tries to evade the pursuit of one or more helicopter cops, who send distance probes from anywhere on the graph. In this paper, we attempt to propose two useful constructions for general problems in this game: a state variable that describes the available game information for the cops, and a Cop Strategy Graph construction that presents all possibilities of the game given a deterministic cop strategy. Then we will use them, along with algorithms and pseudo-code, to explain the relationship between two graph parameters, the localization number and the subdivision number. Researchers have shown that the later has a linear relationship with the former, while the other direction does not. We will revisit their proofs, consolidate the essential correspondence between the two numbers via our proposed constructions, and show an explicit result for the non-linear relationship.