  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.
