THE UPPER FORCING STEINER NUMBER OF A GRAPH
A set is called a Steiner set of Gif every vertex of Gis contained in a Steiner W-tree of G. The Steiner number of Gis the minimum cardinality of its Steiner sets and any Steiner set of cardinality is a minimum Steiner set of G. For a minimum Steiner set Wof G, a subset is called a forcing subset for Wif Wis the unique minimum Steiner set containing T. A forcing subset for Wof minimum cardinality is a minimum forcing subset of W. The forcing Steiner number of W, denoted by is the cardinality of a minimum forcing subset of W. The forcing Steiner number of G, denoted by is where the minimum is taken over all minimum Steiner sets Win G. The upper forcing Steiner number of G, denoted by is where the maximum is taken over all minimum Steiner sets Win G. Some general properties satisfied by this concept are studied. The upper forcing Steiner numbers of certain classes of graphs is studied. Connected graphs of order pwith upper forcing Steiner numbers 0 and 1 are characterized. The necessary and sufficient conditions for to be are given. It is shown for every pair a, bof integers with and there exists a connected graph Gsuch that and
Steiner distance, Steiner number, forcing Steiner number, upper forcing Steiner number.