Discussiones Mathematicae Graph Theory 31(1) (2011) 171-181
doi: 10.7151/dmgt.1536

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THE FORCING STEINER NUMBER OF A GRAPH

A.P. Santhakumaran

Research Department of Mathematics
St. Xavier's College (Autonomous)
Palayamkottai - 627 002, India e-mail: apskumar1953@yahoo.co.in

J. John

Department of Mathematics
Alagappa Chettiar Govt. College of Engineering & Technology
Karaikudi - 630 004, India
e-mail: johnramesh1971@yahoo.co.in

Abstract

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.

Keywords: geodetic number, Steiner number, forcing geodetic number, forcing Steiner number.

2010 Mathematics Subject Classification: 05C12.

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Received 18 February 2009
Revised 24 April 2009
Accepted 27 April 2009