Geodesic
The edge geodetic number and Cartesian product of graphs
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 55-73.

Voir la notice de l'article provenant de la source Library of Science

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).
Keywords: geodetic number, edge geodetic number, linear edge geodetic set, perfect edge geodetic set, (edge, vertex)-geodetic set, superior edge geodetic set 
@article{DMGT_2010_30_1_a4,
     author = {Santhakumaran, A. and Ullas Chandran, S.},
     title = {The edge geodetic number and {Cartesian} product of graphs},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {55--73},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a4/}
}
TY  - JOUR
AU  - Santhakumaran, A.
AU  - Ullas Chandran, S.
TI  - The edge geodetic number and Cartesian product of graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2010
SP  - 55
EP  - 73
VL  - 30
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a4/
LA  - en
ID  - DMGT_2010_30_1_a4
ER  - 
%0 Journal Article
%A Santhakumaran, A.
%A Ullas Chandran, S.
%T The edge geodetic number and Cartesian product of graphs
%J Discussiones Mathematicae. Graph Theory
%D 2010
%P 55-73
%V 30
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a4/
%G en
%F DMGT_2010_30_1_a4
Santhakumaran, A.; Ullas Chandran, S. The edge geodetic number and Cartesian product of graphs. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 55-73. http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a4/

[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.

[2] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990).

[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.

[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.

[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.