On the metric dimension of incidence graph of M\"obius planes

TL;DR

This paper investigates the metric dimension and split-resolving sets of the point-circle incidence graph of M"obius planes, providing bounds and properties related to their sizes and structures.

Contribution

It establishes bounds on the metric dimension, split-resolving sets, and blocking sets of M"obius planes, advancing understanding of their combinatorial properties.

Findings

Metric dimension of M"obius plane of order q is approximately 2q.

Optimal split-resolving set size is between 5q and 2.5q log q.

Smallest blocking set has at most 2q(1 + log(q + 1)) points.

Abstract

We study the metric dimension and optimal split-resolving sets of the point-circle incidence graph of a M\"obius plane. We prove that the metric dimension of a M\"obius plane of order $q$ is around $2q$, and that an optimal split-resolving set has cardinality between approximately $5q$ and $2.5q\log q$. We also prove that a smallest blocking set of a M\"obius plane of order $q$ has at most $2q(1 + \log(q + 1))$ points.