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Discussiones Mathematicae Graph Theory 31(4) (2011) 611-624
DOI: https://doi.org/10.7151/dmgt.1569

On the Forcing Geodetic and Forcing Steiner Numbers of a Graph

A.P. Santhakumaran Research Department of Mathematics St. Xavier's College (Autonomous) Palayamkottai-627 002, India

J. John Department of Mathematics Government College of Engineering Tirunelveli -- 627 007, India

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. The geodetic number g(G) and the forcing geodetic number f(G) of a graph G are defined in [2]. It is proved in [6] that there is no relationship between the geodetic number and the Steiner number of a graph so that there is no relationship between the forcing geodetic number and the forcing Steiner number of a graph. We give realization results for various possibilities of these four parameters.

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

2010 Mathematics Subject Classification: 05C12.

References

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Received 8 June 2009
Revised 23 July 2010
Accepted 28 July 2010