Geodesic
On partial cubes and graphs with convex intervals
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 537-545.

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A graph is called a partial cube if it admits an isometric embedding into a hypercube. Subdivisions of wheels are considered with respect to such embeddings and with respect to the convexity of their intervals. This allows us to answer in negative a question of Chepoi and Tardif from 1994 whether all bipartite graphs with convex intervals are partial cubes. On a positive side we prove that a graph which is bipartite, has convex intervals, and is not a partial cube, always contains a subdivision of $K_4$.
Classification : 05C12, 05C75
Keywords: isometric embeddings; hypercubes; partial cubes; convex intervals; subdivisions 
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Brešar, Boštjan; Klavžar, Sandi. On partial cubes and graphs with convex intervals. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 537-545. http://geodesic.mathdoc.fr/item/CMUC_2002__43_3_a14/