On the Steiner 2-edge connected subgraph polytope

In this paper, we study the Steiner 2–edge connected subgraph polytope. We introduce a large class of valid inequalities for this polytope called the generalized Steiner F–partition inequalities, that generalizes the so–called Steiner F–partition inequalities. We show that these inequalities together with the trivial and the Steiner cut inequalities completely describe the polytope on a class of graphs that generalizes the wheels. We also describe necessary conditions for these inequalities to be facet defining, and as a consequence, we obtain that the separation problem over the Steiner 2–edge connected subgraph polytope for that class of graphs can be solved in polynomial time. Moreover, we discuss that polytope in the graphs that decompose by 3–edge cutsets. And we show that the generalized Steiner F–partition inequalities together with the trivial and the Steiner cut inequalities suffice to describe the polytope in a class of graphs that generalizes the class of Halin graphs when the terminals have a particular disposition. This generalizes a result of Barahona and Mahjoub for Halin graphs. This also yields a polynomial time cutting plane algorithm for the Steiner 2–edge connected subgraph problem in that class of graphs.