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 ) = a , 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].