Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

In a graph Γ = (V,E) a vertex v is resolved by a vertex-set S = {v1, . . . , vn} if its (ordered) distance list with respect to S, (d(v, v1), . . . , d(v, vn)), is unique. A set A ⊂ V is resolved by S if all its elements are resolved by S. S is a resolving set in Γ if it resolves V . The metric dimension of Γ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class. We show that the metric dimension of the incidence graph of a finite projective plane of order q > 23 is 4q − 4, and describe all resolving sets of that size. Let τ2 denote the size of the smallest double blocking set in PG(2, q), the Desarguesian projective plane of order q. We prove that for a semi-resolving set S in the incidence graph of PG(2, q), |S| > min{2q + q/4 − 3, τ2 − 2} holds. In particular, if q > 9 is a square, then the smallest semi-resolving set in PG(2, q) has size 2q + 2 √ q. As a corollary, we get that a blocking semioval in PG(2, q), q > 4, has at least 9q/4 − 3 points.