  Given an undirected graph, the non-empty subgraph polytope is the convex hull
of the characteristic vectors of pairs (F, S) where S is a non-empty subset of
nodes and F is a subset of the edges with both endnodes in S. We obtain a
strong relationship between the non-empty subgraph polytope and the spanning
forest polytope. We further show that these polytopes provide polynomial size
extended formulations for independence polytopes of count matroids, which
generalizes recent results obtained by Iwata et al. referring to sparsity
matroids. As a byproduct, we obtain new lower bounds on the extension
complexity of the spanning forest polytope in terms of extension complexities
of independence polytopes of these matroids.
