Abstract. A graph is radially maximal if its radius decreases after the addition of any edge of its complement. Obviously, for every r there is a radially maximal graph of radius r, as can be shown by complete graphs (in the case r=1) and even cycles (in the case r>1). However, both complete graphs and even cycles are selfcentric graphs, i.e., their radius equals their diameter. One may expect that a graph is radially-maximal if it is either a very dense graph or a balanced (highly symmetric) one. Therefore, it is interesting that for r>2 there are non-selfcentric radially maximal graphs of radius r which are planar. We have the following conjecture:

Conjecture. A non-selfcentric radially maximal graph of radius r>2 on the minimum number of vertices is planar, has maximum degree 3 and the minimum degree 1 and exactly 3r-1 vertices.

First, we prove that for every r>2 there is a radially maximal graph satisfying all the conclusions of the conjecture. Second, using a recent result of Hrnciar, Haviar and Monoszova we prove that the bound for the number of vertices in the conjecture is correct for r=4.