Order and radius of 3-connected graphs (Q2469298)

In a 1999 paper, \textit{Y. Egawa} and \textit{K. Inoue} [Ars. Comb. 51, 89--95 (1999; Zbl 0977.05040)] showed that if \(k\) is an integer with \(k \geq 2\) and \(G\) is a \((2k-1)\)-connected graph with radius \(r\), then \(| V(G)| \geq 2kr - (2k+9)\). The author proves that in the particular case when \(G\) is a 3-connected graph with radius \(r\), \(| V(G)| \geq 4r - 4\). The bound is sharp for \(r \geq 3\). The proof rests on an analysis of the sets \(X_i = \{ w : w\in V(G), d_G(z,w) = i \}\), where \(z\) is a vertex of the center, for \(0 \leq i \leq r\). Since \(| V(G)| = \sum _{i=0} ^r | X_i| \), the result follows when the author is able to show that ''the average of the \(| X_i| \) is only slightly less than 4, if it is less than 4.'' This is accomplished only after an exhausting examination (24 pages!) to exhaust the cases that arise.