Geodesic
The strong isometric dimension of finite reflexive graphs
Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 1, pp. 23-38

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The strong isometric dimension of a reflexive graph is related to its injective hull: both deal with embedding reflexive graphs in the strong product of paths. We give several upper and lower bounds for the strong isometric dimension of general graphs; the exact strong isometric dimension for cycles and hypercubes; and the isometric dimension for trees is found to within a factor of two.
Keywords: isometric, embedding, strong product, injective hull, paths, distance, metric 
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Fitzpatrick, Shannon; Nowakowski, Richard. The strong isometric dimension of finite reflexive graphs. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 1, pp. 23-38. http://geodesic.mathdoc.fr/item/DMGT_2000_20_1_a1/