The minimum weakly connected independent set problem: Polyhedral results and Branch-and-Cut

Let G=(V,E) be a connected graph. An independent set W⊂V is said to be weakly connected if the spanning subgraph GW=(V,δ(W)) is connected where δ(W) is the set of edges with exactly one end in W. We present an integer programming formulation for the minimum weakly connected independent set problem and discuss its associated polytope. Some classical graph operations are also used for defining new polyhedra from pieces. We give valid inequalities and describe heuristic separation procedures. Finally, we develop a Branch-and-Cut algorithm and test it on two classes of graphs.