The Edge-to-Edge Steiner Number of a Graph

Abstract

For a non-empty set of edges in a connected graph,the edge-to-vertex Steiner distance of is the minimum size of a tree containing  and is called an edge-to-vertex Steiner tree with respect to  or a Steiner  - tree.A set  is called an edge-to-edge Steiner set if every edge of  lies on a Steiner  - tree of . The edge-to-edge Steiner number of is the minimum cardinality of its edge-to-edge Steiner sets and any edge-to-edge Steiner sets of cardinality is called a minimum edge-to-edge Steiner set of  or a -set of . The edge-to-edge Steiner number of certain classes of graphs are determined. We characterize a connected graph of size with an edge-to-edge Steiner number or .