The Vertex Steiner Number of a Graph

Abstract

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 s_x (G). Some general properties satisfied by this concept 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 s_x (G)= b for some vertex x in G, where s(G) is the Steiner number of a graph.