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

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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).
Keywords: geodetic number, edge geodetic number, linear edge geodetic set, perfect edge geodetic set, (edge, vertex)-geodetic set, superior edge geodetic set 
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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/