Partial cubes and their ττ-graphs

The ττ-graph GτGτ of a partial cube GG has the equivalence classes of the Djoković–Winkler relation as vertices, two classes EE and FF being adjacent if some edges e∈Ee∈E and f∈Ff∈F induce a convex P3P3. It is shown that for every graph GG there exists a median graph MM such that G=MτG=Mτ, that GτGτ is connected if and only if GG is a Cartesian prime graph, and that for a median graph GG its ττ-graph is KnKn-free if and only if GG contains no convex K1,nK1,n.