The vertex steiner number of a graph

Let x be a vertex of a connected graph G and W⊂V(G) such that x∉W. Then W is called an x-Steiner set of extit{G} if W∪{x} is a Steiner set of extit{G}. The minimum cardinality of an x- extit{Steiner set} of extit{G} is defined as x- extit{Steiner number} of extit{G} and denoted by sx(G). Some general properties satisfied by these concepts are studied. The x- extit{Steiner numbers} of certain classes of graphs are determined. Connected graphs of order extit{p} with x-Steiner number 1 or p−1 are characterized. It is shown that for every pair extit{a}, extit{b} of integers with 2≤a≤b, there exists a connected graph extit{G} such that s(G)} = a and sx(G)=b for some vertex x in extit{G}, where s(G) is the extit{Steiner number} of a graph.