The diameter and radius of radially maximal graphs
Abstract
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 d 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 d 2r-2, then there exists a radially maximal graph with radius r and diameter d. We prove this conjecture and a little more.
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