Critical extreme points of the 2-edge connected spannning subgraph polytope

In this paper we study the extreme paints of the polytope P(G), the linear relaxation of the 2-edge connected spanning subgraph polytope of a graph G. We introduce a partial ordering on the extreme points of P(G) and give necessary conditions for a non-integer extreme point of P(G) to be minimal with respect to that ordering. We show that, if x is a non-integer minimal extreme point of P(G), then G and x can be reduced, by means of some reduction operations, to a graph G' and an extreme point x' of P(G') where G' and x' satisfy some simple properties. As a consequence we obtain a characterization of the perfectly 2-edge connected graphs, the graphs for which the polytope P(G) is integral.