Geodesic
Products of Geodesic Graphs and the Geodetic Number of Products
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 35-42

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Given a connected graph and a vertex x ∈ V (G), the geodesic graph P_x(G) has the same vertex set as G with edges uv iff either v is on an x − u geodesic path or u is on an x − v geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of K_m,n with itself, where m, n ≥ 4, is equal to the minimum of m, n and eight.
Keywords: geodesic graph, geodetic number, Cartesian products 
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Soloff, Jake A.; Márquez, Rommy A.; Friedler, Louis M. Products of Geodesic Graphs and the Geodetic Number of Products. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 35-42. http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a2/