Generalizing the Single-Crossing Property on Lines and Trees to Intermediate Preferences on Median Graphs

Demange (2012) generalized the classical single-crossing property to the intermediate property on median graphs and proved that the representative voter theorem still holds for this more general framework. We complement her result with proving that the linear orders of any profile which is intermediate on a median graph  form a Condorcet domain. We prove that for any median graph there exists a profile that is intermediate with respect to that graph and that one may need at least as many alternatives as vertices to construct such a profile.  We provide a polynomial-time algorithm to recognize whether or not a given profile is intermediate with respect to some median graph. Finally, we show that finding winners for the Chamberlin-Courant rule is polynomial-time solvable  for profiles that are single-crossing on a tree.