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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1536,
author = {A.P. Santhakumaran and J. John},
title = {The forcing steiner number of a graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {171-181},
zbl = {1284.05085},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1536}
}
A.P. Santhakumaran; J. John. The forcing steiner number of a graph. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 171-181. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1536/
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