On the p-median polytope of fork-free graphs

We study a prize collecting version of the uncapacitated facility location problem and of the p-median problem. We say that the uncapacitated facility location polytope has the intersection property, if adding the extra equation that fixes the number of opened facilities does not create any fractional extreme point. We show that this property holds if and only if the graph has no fork. A fork is a particular subgraph. We give a complete description of the polytope for this class of graphs.