Geodesic
On θ-graphs of partial cubes
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 2, pp. 313-321

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The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K₁ by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ-graphs.
Keywords: intersection graph, partial cube, median graph, expansion theorem, Cartesian product of graphs 
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Klavžar, Sandi; Kovse, Matjaz. On θ-graphs of partial cubes. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 2, pp. 313-321. http://geodesic.mathdoc.fr/item/DMGT_2007_27_2_a7/