I am struggling with the following question. Given an undirected graph $G = (V,E)$, the edge cover polyope $EC(G) \subseteq \mathbb{R^{|E|}}$ is the convex hull of the incidence vectors of edge covers of $G$.
Here is a Linear Relaxation of the edge cover polytope, which I am trying to show is integral:
(1) For each $v \in V, \sum_{e \in \delta(v)} x_e \geq 1.$
(2) For each $S \subseteq V, |S|$ is odd, $\sum_{e \in E[S] \cup \delta(S)} x_e \geq \frac{|S| + 1}{2}.$
This is a known result, but I cannot find a proof of this fact anywhere except in Schrijver's book. But that is a complicated proof that shows that the polytope is TDI. I just want a proof that it is integral. Can anyone point me to some source for the proof?
How can this problem be reduced to the integrality of the perfect matching polytope?