Edge fixed Steiner number of a graph

For a non-empty set W of vertices in a connected graph G, the Steiner distance d(W) of W is the minimum size of a connected subgraph of G containing W. Necessarily, each such subgraph is a tree and is called a Steiner tree with respect to W or a Steiner W-tree. S(W) denotes the set of vertices that lies in Steiner W-trees. Let G be a connected graph with at least 2 vertices. A set W ⊆ V(G) is called a Steiner set of G if S(W) = V(G). The Steiner number s(G) is the minimum cardinality of a Steiner set. Let G be a connected graph with at least 3 vertices. For an edge e = xy in G, a set W ⊆ V(G) − {x, y} is called an edge fixed Steiner set of G if W’ = W ∪ {x, y} is a Steiner set of G. The minimum cardinality of an edge fixed Steiner set is called the edge fixed Steiner number of G and is denoted by se(G). Also the Steiner W-tree necessarily contains the edge e and is called edge fixed Steiner W-tree. In this paper, we begin with an investigation of this parameter. 772 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi