Resolving sets and semi-resolving sets in finite projective planes

TL;DR

This paper determines the metric dimension of finite projective planes for large q, characterizes resolving sets, and establishes bounds on semi-resolving sets, including exact sizes for certain q.

Contribution

It provides the exact metric dimension for projective planes of order q ≥ 23 and bounds on semi-resolving sets, including their minimal sizes in specific cases.

Findings

Metric dimension of PG(2,q) is 4q-4 for q ≥ 23.

Minimal semi-resolving set size in PG(2,q) is 2q+2√q for q a square ≥ 9.

Bounds involving double blocking sets are established.

Abstract

We show that the metric dimension of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in $\mathrm{PG}(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of $\mathrm{PG}(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in $\mathrm{PG}(2,q)$ has size $2q+2\sqrt{q}$.