Three quotient graphs factored through the Djokovi´ c-Winkler relation Sandi Klavˇ zar ∗ Department of Mathematics and Computer Science PeF, University of Maribor Koroˇ ska cesta 160, 2000 Maribor, Slovenia email: sandi.klavzar@uni-mb.si Matjaˇ z Kovˇ se Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia email: matjaz.kovse@uni-mb.si Abstract For a partial cube (that is, an isometric subgraph of a hypercube) G, quo- tient graphs G # , G τ , and G ∼ have the equivalence classes of the Djokovi´ c- Winkler relation as the vertex set, while edges are defined in three different natural ways. Several results on these quotients are proved and the con- cepts are compared. For instance, for every graph G there exists a median graph M such that G = M τ . Triangle-free and complete quotient graphs are treated and it is proved that for a median graph G its τ -graph is triangle-free if and only if G contains no convex K 1,3 . Connectedness and the question of when quotients yield the same graphs are also treated. Key words: hypercube, isometric subgraph, partial cube, median graph, Carte- sian product of graphs AMS subject classification (2000): 05C75, 05C12 1 Introduction The celebrated Djokovi´ c-Winkler relation Θ [7, 20], is defined on the edge set of a graph G in the following way. Edges xy and uv of G are in relation Θ if d(x, u)+ d(y,v) = d(x, v)+ d(y,u) . * Supported by the Ministry of Science of Slovenia under the grant P1-0297. 1