THE RESTRAINED STEINER NUMBER OF A GRAPH

For a connected graph G = (V, E) of order p, 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. A set W of vertices of a graph G is a restrained Steiner set if W is a Steiner set, and if either W = V or the subgraph G[V − W ] induced by V − W has no isolated vertices. The minimum cardinality of a restrained Steiner set of G is the restrained Steiner number of G, and is denoted by s r (G). The restrained Steiner number of certain classes of graphs are determined. Connected graphs of order p with restrained Steiner number 2 are characterized. Various necessary conditions for the restrained Steiner number of a graph to be p are given. It is shown that, for integers a, b and p with 4 ≤ a ≤ b ≤ p, there exists a connected graph G of order p such that s(G) = a and s r (G) = b. It is also proved that for every pair of integers a, b with a ≥ 3 and b ≥ 3, there exists a connected graph G with s r (G) = a and g r (G) = b