Resolving sets and semi-resolving sets in finite projective planes

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 $\mathrm{P}\mathrm{G}(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of $\mathrm{P}\mathrm{G}(2,q)$, $\mathrm{\mid }S\mathrm{\mid }\ge \min \{2q+q\mathrm{/}4-3,{\tau }_2-2\}$ holds. In particular, if $q\ge 9$ is a square, then the smallest semi-resolving set in $\mathrm{P}\mathrm{G}(2,q)$ has size $2q+2\sqrt{q}$.