Resolving sets and semi-resolving sets in finite projective planes

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 $2q+2\sqrt{q}$.