Lovász-Schrijver SDP-operator, near-perfect graphs and near-bipartite graphs

We study the Lovász-Schrijver lift-and-project operator (LS+) based on the cone of symmetric, positive semidefinite matrices, applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the LS+-operator generates the stable set polytope in one step has been open since 1990. We call these graphs LS+-perfect. In the current contribution, we pursue a full combinatorial characterization of LS+perfect graphs and make progress towards such a characterization by establishing a new, close relationship among LS+-perfect graphs, near-bipartite graphs and a newly introduced concept of full-support-perfect graphs.