  A graph is called radially maximal if it is not complete and the addition of
any new edge decreases its radius. In 1976 Harary and Thomassen proved that the
radius $r$ and diameter $d$ of any radially maximal graph satisfy $r\le d\le
2r-2.$ Dutton, Medidi and Brigham rediscovered this result with a different
proof in 1995 and they posed the conjecture that the converse is true, that is,
if $r$ and $d$ are positive integers satisfying $r\le d\le 2r-2,$ then there
exists a radially maximal graph with radius $r$ and diameter $d.$ We prove this
conjecture and a little more.
