A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. Harary and Thomassen [‘Anticritical graphs’, Math. Proc. Cambridge Philos. Soc.79(1) (1976), 11–18] proved that the radius r and diameter d of any radially maximal graph satisfy $r\le d\le 2r-2.$ Dutton et al. [‘Changing and unchanging of the radius of a graph’, Linear Algebra Appl.217 (1995), 67–82] rediscovered this result with a different proof and conjectured 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.

中文翻译：

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径向最大图形的直径和半径

如果图不完整并且添加任何新边会减小其半径，则该图称为径向极大图。Harary 和 Thomassen ['反批判图'，数学。过程。剑桥哲学。社会党。79(1) (1976), 11-18] 证明了半径r和直径d任何径向极大图的满足$r\le d\le 2r-2.$达顿等。['图形半径的变化和不变',线性代数应用程序。217(1995), 67-82] 用不同的证明重新发现了这个结果，并推测反之亦然，也就是说，如果r和d是满足的正整数$r\le d\le 2r-2,$那么存在一个半径最大的图r和直径$d.$我们证明了这个猜想和更多。