Abstract. A graph G is radially-maximal if adding of any edge from its complement decreases its radius. 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). Both complete graphs and 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 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. Such graphs are neither symmetric nor dense. In one of the previous papers we have the following conjecture:

Conjecture. Let G be a non-selfcentric radially-maximal graph of radius r>2 on the minimum number of vertices. Then
(a) G has exactly 3r-1 vertices;
(b) G is planar;
(c) the minimum degree of G is 1 and its maximum degree is 3.

Using an enhanced exhaustive computer search we prove this conjecture for r=4,5. In the following table, for each value of r, 3≤r≤5, we give the number of mutually nonisomorphic graphs satisfying the Conjecture (the numbers for r=3,4 were known also before).

For r=3 there are 2 graphs;
for r=4 there are 8 graphs;
for r=5 there are 22 graphs.