# The edge geodetic number and Cartesian product of graphs

A.P. Santhakumaran; S.V. Ullas Chandran

Discussiones Mathematicae Graph Theory (2010)

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

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topA.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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- [2] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). Zbl0688.05017
- [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] G. Chartrand and P. Zhang, Introduction to Graph Theory (Tata McGraw-Hill Edition, New Delhi, 2006).
- [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] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley-Interscience, New York, 2000).
- [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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