# The forcing steiner number of a graph

• Volume: 31, Issue: 1, page 171-181
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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A.P. Santhakumaran, and J. John. "The forcing steiner number of a graph." Discussiones Mathematicae Graph Theory 31.1 (2011): 171-181. <http://eudml.org/doc/270960>.

@article{A2011,
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.},
author = {A.P. Santhakumaran, J. John},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {geodetic number; Steiner number; forcing geodetic number; forcing Steiner number},
language = {eng},
number = {1},
pages = {171-181},
title = {The forcing steiner number of a graph},
url = {http://eudml.org/doc/270960},
volume = {31},
year = {2011},
}

TY - JOUR
AU - A.P. Santhakumaran
AU - J. John
TI - The forcing steiner number of a graph
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 1
SP - 171
EP - 181
AB - 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.
LA - eng
KW - geodetic number; Steiner number; forcing geodetic number; forcing Steiner number
UR - http://eudml.org/doc/270960
ER -

## References

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6. [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. Zbl0825.68490
7. [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. Zbl1062.05052
8. [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. [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. Zbl1200.05073
10. [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. Zbl1175.05074

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