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Condorcet domains are sets of linear orders with the property that, wheneverthe preferences of all voters belong to this set, the majority relation has nocycles. We observe that, without loss of generality, such domain can be assumedto be closed in the sense that it contains the majority relation of everyprofile with an odd number of individuals whose preferences belong to thisdomain.  We show that every closed Condorcet domain is naturally endowed with thestructure of a median graph and that, conversely, every median graph isassociated with a closed Condorcet domain (which may not be a unique one). Thesubclass of those Condorcet domains that correspond to linear graphs (chains)are exactly the preference domains with the classical single crossing property.As a corollary, we obtain that the domains with the so-called `representativevoter property' (with the exception of a 4-cycle) are the single crossingdomains.  Maximality of a Condorcet domain imposes additional restrictions on theunderlying median graph. We prove that among all trees only the chains caninduce maximal Condorcet domains, and we characterize the single crossingdomains that in fact do correspond to maximal Condorcet domains.  Finally, using Nehring's and Puppe's (2007) characterization of monotoneArrowian aggregation, our analysis yields a rich class of strategy-proof socialchoice functions on any closed Condorcet domain.

Condorcet Domains, Median Graphs and the Single Crossing Property