Two-edge-connected (not necessarily spanning) subgraphs and polyhedra

Given a graph $G$, we study the $2$-edge-connected subgraph polytope $\mathrm{T}\mathrm{E}\mathrm{C}\mathrm{S}\mathrm{P}(G)$, which is given by the convex hull of the incidence vectors of all $2$-edge-connected subgraphs of $G$. We describe the lattice points of this polytope by linear inequalities which provides an ILP-algorithm for finding a $2$-edge-connected subgraph of maximum weight. Furthermore, we characterize when these inequalities define facets of $\mathrm{T}\mathrm{E}\mathrm{C}\mathrm{S}\mathrm{P}(G)$. We also consider further types of supporting hyperplanes of $\mathrm{T}\mathrm{E}\mathrm{C}\mathrm{S}\mathrm{P}(G)$ and study when they are facet-defining. Finally, we investigate the efficiency of our considered inequalities practically on some classes of graphs.