# On the forcing geodetic and forcing steiner numbers of a graph

• Volume: 31, Issue: 4, page 611-624
• 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. 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.

## How to cite

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A.P. Santhakumaran, and J. John. "On the forcing geodetic and forcing steiner numbers of a graph." Discussiones Mathematicae Graph Theory 31.4 (2011): 611-624. <http://eudml.org/doc/271013>.

@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. 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.},
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 = {4},
pages = {611-624},
title = {On the forcing geodetic and forcing steiner numbers of a graph},
url = {http://eudml.org/doc/271013},
volume = {31},
year = {2011},
}

TY - JOUR
AU - A.P. Santhakumaran
AU - J. John
TI - On the forcing geodetic and forcing steiner numbers of a graph
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 4
SP - 611
EP - 624
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. 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.
LA - eng
KW - geodetic number; Steiner number; forcing geodetic number; forcing Steiner number
UR - http://eudml.org/doc/271013
ER -

## References

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1. [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). Zbl0688.05017
2. [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. Zbl0927.05025
3. [3] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39 (2002) 1-6, doi: 10.1002/net.10007. Zbl0987.05047
4. [4] G. Chartrand and P. Zhang, The Steiner number of a graph, Discrete Math. 242 (2002) 41-54, doi: 10.1016/S0012-365X(00)00456-8. Zbl0988.05034
5. [5] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17 (1993) 89-95, doi: 10.1016/0895-7177(93)90259-2. Zbl0825.68490
6. [6] I.M. Pelayo, Comment on 'The Steiner number of a graph' by G. Chartrand and P. Zhang, Discrete Math. 242 (2002) 41-54.
7. [7] A.P. Santhakumaran, P. Titus and J. John, On the connected geodetic number of a graph, J. Combin. Math. Combin. Comput. 69 (2009) 219-229. Zbl1200.05073
8. [8] A.P. Santhakumaran, P. Titus and J. John, The upper connected geodetic number and forcing connected geodetic number of a graph, Discrete Appl. Math. 159 (2009) 1571-1580, doi: 10.1016/j.dam.2008.06.005. Zbl1175.05074

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