On the metric dimension of affine planes, biaffine planes and generalized quadrangles

In this paper the metric dimension of (the incidence graphs of) particular partial linear spaces is considered. We prove that the metric dimension of an affine plane of order q ≥ 13 is 3q − 4 and describe all resolving sets of that size if q ≥ 23. The metric dimension of a biaffine plane (also called a flag-type elliptic semiplane) of order q ≥ 4 is shown to fall between 2q−2 and 3q−6, while for Desarguesian biaffine planes the lower bound is improved to 8q/3− 7 under q ≥ 7, and to 3q− 9√q under certain stronger restrictions on q. We determine the metric dimension of generalized quadrangles of order (s, 1), s arbitrary. We derive that the metric dimension of generalized quadrangles of order (q, q), q ≥ 2, is at least max{6q−27, 4q−7}, while for the classical generalized quadrangles W (q) and Q(4, q) it is at most 8q.