The forcing Steiner number of a graph

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steiner set containing T . A forcing subset for W of minimum cardinality is a minimum forcing subset of W . The forcing Steiner number of W , denoted by fs(W ), is the cardinality of a minimum forcing subset of W . The forcing Steiner number of G, denoted by fs(G), is fs(G) = min{fs(W )}, where the minimum is taken over all minimum Steiner sets W in G. Some general properties satisfied by this concept are studied. The forcing Steiner numbers of certain classes of graphs are determined. It is shown for every pair a, b of integers with 0 ≤ a < b, b ≥ 2, there exists a connected graph G such that fs(G) = a and s(G) = b.