Geodetic sets and Steiner sets in graphs

Suppose that G is a simple graph. Let g(G) and s(G) be the geodetic number and the Steiner number of G, respectively. In this note, we prove that, for any nonnegative integers a, b and r with a >= b >= 3 and r >= 3, there exists a connected graph G with g(G) = b. s(G) = 0, and r(G) = d(G) = r where r(G) and d(G) are the radius and diameter of G, respectively. This result answers the conjecture of Chartrand and Zhang [G. Chartrand, P. Zhang, The Steiner number of a graph, Discrete Math. 242 (2002) 41-54]. (C) 2008 Elsevier B.V. All rights reserved.