The localization number of designs

TL;DR

This paper investigates the localization number of incidence graphs of combinatorial designs, providing bounds and exact values for specific classes like projective and affine planes, advancing understanding of cops and robbers game on these graphs.

Contribution

It offers new bounds and exact values for the localization number of incidence graphs of various combinatorial designs, including projective and affine planes.

Findings

Exact localization numbers for projective and affine planes.

Bounds for Steiner systems and transversal designs.

Insights into the localization game on design-based graphs.

Abstract

We study the localization number of incidence graphs of designs. In the localization game played on a graph, the cops attempt to determine the location of an invisible robber via distance probes. The localization number of a graph $G$, written $\zeta(G)$, is the minimum number of cops needed to ensure the robber's capture. We present bounds on the localization number of incidence graphs of balanced incomplete block designs. Exact values of the localization number are given for the incidence graphs of projective and affine planes. Bounds are given for Steiner systems and for transversal designs.