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. 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.
2017 ◽
Vol 97
(1)
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pp. 15-25
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2017 ◽
Vol 38
(8)
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pp. 3101-3144
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2016 ◽
Vol Vol. 17 no. 3
(Combinatorics)
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