The diameter and radius of radially maximal graphs

TL;DR

This paper proves a conjecture that for any positive integers r and d satisfying r ≤ d ≤ 2r-2, there exists a radially maximal graph with radius r and diameter d, confirming a longstanding hypothesis.

Contribution

The paper proves the conjecture that all pairs of radius and diameter satisfying r ≤ d ≤ 2r-2 correspond to radially maximal graphs, extending previous results.

Findings

Confirmed the existence of radially maximal graphs for all valid (r, d) pairs.

Extended the known bounds relating radius and diameter in radially maximal graphs.

Provided a constructive proof for the conjecture.

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 $r\le d\le 2r-2.$ 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.