# The edge geodetic number and Cartesian product of graphs

• Volume: 30, Issue: 1, page 55-73
• ISSN: 2083-5892

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## Abstract

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For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g₁(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several classes of graphs. Also we obtain a necessary condition of G for which g₁(G ☐ K₂) = g₁(G).

## How to cite

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A.P. Santhakumaran, and S.V. Ullas Chandran. "The edge geodetic number and Cartesian product of graphs." Discussiones Mathematicae Graph Theory 30.1 (2010): 55-73. <http://eudml.org/doc/270858>.

@article{A2010,
abstract = {For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g₁(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several classes of graphs. Also we obtain a necessary condition of G for which g₁(G ☐ K₂) = g₁(G).},
author = {A.P. Santhakumaran, S.V. Ullas Chandran},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {geodetic number; edge geodetic number; linear edge geodetic set; perfect edge geodetic set; (edge, vertex)-geodetic set; superior edge geodetic set},
language = {eng},
number = {1},
pages = {55-73},
title = {The edge geodetic number and Cartesian product of graphs},
url = {http://eudml.org/doc/270858},
volume = {30},
year = {2010},
}

TY - JOUR
AU - A.P. Santhakumaran
AU - S.V. Ullas Chandran
TI - The edge geodetic number and Cartesian product of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 1
SP - 55
EP - 73
AB - For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g₁(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several classes of graphs. Also we obtain a necessary condition of G for which g₁(G ☐ K₂) = g₁(G).
LA - eng
KW - geodetic number; edge geodetic number; linear edge geodetic set; perfect edge geodetic set; (edge, vertex)-geodetic set; superior edge geodetic set
UR - http://eudml.org/doc/270858
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, (2007), Discrete Math. 308 (2008) 5555-5561, doi: 10.1016/j.disc.2007.10.007. Zbl1200.05060
2. [2] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). Zbl0688.05017
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, Introduction to Graph Theory (Tata McGraw-Hill Edition, New Delhi, 2006).
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] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley-Interscience, New York, 2000).
7. [7] A.P. Santhakumaran and J. John, Edge geodetic number of a graph, J. Discrete Math. Sciences & Cryptography 10 (2007) 415-432. Zbl1133.05028

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