The edge Steiner number of a graph
A. P. Santhakumaran, J. John · Journal of Discrete Mathematical Sciences and Cryptography · 2007
For a non-trivial connected graph G of order p, a set W⊆V(G) is called an edge Steiner set of G if every edge of G is contained in a Steiner W-tree of G. The edge Steiner number s 1 (G) of G is the minimum cardinality of its edge Steiner sets and any edge Steiner set of cardinality s 1 (G) is a minimum edge Steiner set of G. Connected graphs with edge Steiner number 2 are characterized. Various necessary conditions for the edge Steiner number of a graph to be p–1 or p are given. It is shown that every pair k, p of integers with 2≤k≤p is realizable as the edge Steiner number and order of some connected graph. For positive integers r, d and k≥2 with r