Geodetic sets and Steiner sets in graphs (Q1043600)

Suppose \(G\) is a connected graph and \(u,v\) vertices of \(G\). The geodesic interval \(I[u,v]\) is the collection of all vertices that lie on some shortest \(u\)--\(v\) path in \(G\). If \(S \subseteq V(G)\), then \(I[S] = \bigcup_{u,v \in S} I[u,v]\). The geodetic number \(g(G)\) of \(G\) is the smallest cardinality of a set \(S\) of vertices such that \(I[S] = V(G)\). A Steiner tree for a set \(F\) of vertices is the smallest size of a connected subgraph of \(G\) that contains \(F\). The Steiner interval \(S[F]\) of \(F\) is the collection of all vertices that lie on some Steiner tree for \(F\). A set of vertices is a Steiner set if \(S[F] = V(G)\). The smallest cardinality of a Steiner set, \(s(G)\), is called the Steiner number of \(G\). The author solves a conjecture posed by Chartrand and Zhang [\textit{G. Chartrand} and \textit{P. Zhang}, ``The Steiner number of a graph'', Discrete Math. 242, No.1-3, 41--54 (2002; Zbl 0887.05024)], namely that for given integers \(a \geq b \geq 3\) and \(r \geq 3\), there is a graph with Steiner number \(a\), geodetic number \(b\) and radius and diameter both equal to \(r\).