DMGT

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

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 f_s(W), is the cardinality of a minimum forcing subset of W. The forcing Steiner number of G, denoted by f_s(G), is f_s(G) = min{f_s(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 f_s(G) = a and s(G) = b.

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

2010 Mathematics Subject Classification: 05C12.

References

[1]	F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990).
[2]	G. Chartrand and P. Zhang, The forcing geodetic number of a graph, Discuss. Math. Graph Theory 19 (1999) 45-58, doi: 10.7151/dmgt.1084.
[3]	G. Chartrand and P. Zhang, The forcing dimension of a graph, Mathematica Bohemica 126 (2001) 711-720.
[4]	G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39 (2002) 1-6, doi: 10.1002/net.10007.
[5]	G. Chartrand, F. Harary and P. Zhang, The Steiner Number of a Graph, Discrete Math. 242 (2002) 41-54.
[6]	F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modelling 17 (1993) 89-95, doi: 10.1016/0895-7177(93)90259-2.
[7]	C. Hernando, T. Jiang, M. Mora, I.M. Pelayo and C. Seara, On the Steiner, geodetic and hull numbers of graphs, Discrete Math. 293 (2005) 139-154, doi: 10.1016/j.disc.2004.08.039.
[8]	I.M. Pelayo, Comment on "The Steiner number of a graph" by G. Chartrand and P. Zhang, Discrete Math. 242 (2002) 41-54.
[9]	A.P. Santhakumaran, P. Titus and J. John, On the Connected Geodetic Number of a Graph, J. Combin. Math. Combin. Comput. 69 (2009) 205-218.
[10]	A.P. Santhakumaran, P. Titus and J. John, The Upper Connected Geodetic Number and Forcing Connected Geodetic Number of a Graph, Discrete Appl. Math. 157 (2009) 1571-1580, doi: 10.1016/j.dam.2008.06.005.

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