The Steiner number of a graph Original Research Article

For a connected graph G of order n⩾3 and a set W⊆V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W⊆V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W)=V(G). The minimum cardinality among the Steiner sets of G is the Steiner number s(G). Connected graphs of order n with Steiner number n, n−1, or 2 are characterized. It is shown that every pair k,n of integers with 2⩽k⩽n is realizable as the Steiner number and order of some connected graph. For positive integers r,d, and k⩾2 with r⩽d⩽2r, there exists a connected graph of radius r, diameter d, and Steiner number k. Also, for integers n,d, and k with 2⩽d