Subgraph polytopes and independence polytopes of count matroids

Given a graph, the non-empty subgraph polytope is the convex hull of the characteristic vectors of all pairs (F,S)(F,S) where SS is a non-empty subset of nodes and FF is a subset of the edges with both endnodes in SS. We obtain a strong relationship between non-empty subgraph polytopes and spanning forest polytopes relating their extension complexities. We further show that these polytopes provide polynomial size extended formulations for independence polytopes of count matroids, generalizing recent results on sparsity matroids.