# The geodetic number of strong product graphs

• Volume: 30, Issue: 4, page 687-700
• ISSN: 2083-5892

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## Abstract

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For two vertices u and v of a connected graph G, the set ${I}_{G}\left[u,v\right]$ consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets ${I}_{G}\left[u,v\right]$ for u,v ∈ S is denoted by ${I}_{G}\left[S\right]$. A set S ⊆ V(G) is a geodetic set if ${I}_{G}\left[S\right]=V\left(G\right)$ and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.

## How to cite

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A.P. Santhakumaran, and S.V. Ullas Chandran. "The geodetic number of strong product graphs." Discussiones Mathematicae Graph Theory 30.4 (2010): 687-700. <http://eudml.org/doc/270905>.

@article{A2010,
abstract = {For two vertices u and v of a connected graph G, the set $I_G[u,v]$ consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets $I_G[u,v]$ for u,v ∈ S is denoted by $I_G[S]$. A set S ⊆ V(G) is a geodetic set if $I_G[S] = V(G)$ and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.},
author = {A.P. Santhakumaran, S.V. Ullas Chandran},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {geodetic number; extreme vertex; extreme geodesic graph; open geodetic number; double domination number},
language = {eng},
number = {4},
pages = {687-700},
title = {The geodetic number of strong product graphs},
url = {http://eudml.org/doc/270905},
volume = {30},
year = {2010},
}

TY - JOUR
AU - A.P. Santhakumaran
AU - S.V. Ullas Chandran
TI - The geodetic number of strong product graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 4
SP - 687
EP - 700
AB - For two vertices u and v of a connected graph G, the set $I_G[u,v]$ consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets $I_G[u,v]$ for u,v ∈ S is denoted by $I_G[S]$. A set S ⊆ V(G) is a geodetic set if $I_G[S] = V(G)$ and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.
LA - eng
KW - geodetic number; extreme vertex; extreme geodesic graph; open geodetic number; double domination number
UR - http://eudml.org/doc/270905
ER -

## References

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1. [1] B. Bresar, S. Klavžar and A.T. Horvat, On the geodetic number and related metric sets in Cartesian product graphs, Discrete Math. 308 (2008) 5555-5561, doi: 10.1016/j.disc.2007.10.007. Zbl1200.05060
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4. [4] G. Chartrand, F. Harary, H.C. Swart and P. Zhang, Geodomination in Graphs, Bulletin of the ICA 31 (2001) 51-59. Zbl0969.05048
5. [5] G. Chartrand and P. Zhang, Introduction to Graph Theory (Tata McGraw-Hill Edition, New Delhi, 2006).
6. [6] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Mathl. Comput. Modeling 17 (1993) 89-95, doi: 10.1016/0895-7177(93)90259-2. Zbl0825.68490
7. [7] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213. Zbl0993.05104
8. [8] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley-Interscience, New York, 2000).
9. [9] A.P. Santhakumaran and S.V. Ullas Chandran, The hull number of strong product of graphs, (communicated). Zbl1229.05240

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