# Lov\'{a}sz-Schrijver SDP-operator, near-perfect graphs and near-bipartite graphs

We study the Lov\'{a}sz-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 ${\LS}_+$-\emph{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.

## Citation

arXiv:1411.2069, November 2014