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 g if w ∪ {x} is a steiner set of g. the minimum cardinality of an x-steiner set of g is defined as x-steiner number of g and denoted by sx(g). some general properties satisfied by these concepts are studied. the x-steiner numbers of certain classes of graphs are determined. connected graphs of order p with x-steiner number 1 or p − 1 are characterized. it is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph g such that s(g) = a and sx(g) = b for some vertex x in g, where s(g) is the steiner number of a graph