Resolving sets and semi-resolving sets in finite projective planes

Abstract

We show that the metric dimension of a finite projective plane of order $q\ge 23$ 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 $\mathrm{PG}(2,q)$, $|S|\ge \min \{2q+q/4-3,{\tau }_2-2\}$ holds. In particular, if $q\ge 9$ is a square, then the smallest semi-resolving set in $\mathrm{PG}(2,q)$ has size $2q+2\sqrt{q}$.

arXiv comment21 pages, 3 figures. Version 3 contains clarifications and minor corrections regarding the list and the figure of the 32 types of smallest resolving sets, and a supplementary page explaining these modifications