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).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1476,
author = {A.P. Santhakumaran and S.V. Ullas Chandran},
title = {The edge geodetic number and Cartesian product of graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {30},
year = {2010},
pages = {55-73},
zbl = {1215.05049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1476}
}
A.P. Santhakumaran; S.V. Ullas Chandran. The edge geodetic number and Cartesian product of graphs. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 55-73. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1476/
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