Abstract We study the Lovasz-Schrijver SDP-operator applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the SDP-operator generates the stable set polytope in one step has been open since 1990. In an earlier publication, we named these graphs N + -perfect. In the current contribution, we propose a conjecture on combinatorial characterization of N + -perfect graphs and make progress towards such a full combinatorial characterization by establishing a new, close relationship among N + -perfect graphs, near-bipartite graphs and a newly introduced concept of full-support-perfect graphs.