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.

In this paper we prove the conjecture for unicyclic graphs. Moreover, we characterize unicyclic graphs satisfying this conjecture. Such graphs must have radius at least 5, and we prove that the number of these graphs of radius r is r³/48+O(r²).