@misc{Wagler2002, author = {Wagler, Annegret}, title = {Relaxing Perfectness: Which Graphs are 'Almost' Perfect?}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6700}, number = {02-03}, year = {2002}, abstract = {For all perfect graphs, the stable set polytope STAB\$(G)\$ coincides with the fractional stable set polytope QSTAB\$(G)\$, whereas STAB\$(G) \subset\$ QSTAB\$(G)\$ holds iff \$G\$ is imperfect. Padberg asked in the early seventies for ``almost'' perfect graphs. He characterized those graphs for which the difference between STAB\$(G)\$ and QSTAB\$(G)\$ is smallest possible. We develop this idea further and define three polytopes between STAB\$(G)\$ and QSTAB\$(G)\$ by allowing certain sets of cutting planes only to cut off all the fractional vertices of QSTAB\$(G)\$. The difference between QSTAB\$(G)\$ and the largest of the three polytopes coinciding with STAB\$(G)\$ gives some information on the stage of imperfectness of the graph~\$G\$. We obtain a nested collection of three superclasses of perfect graphs and survey which graphs are known to belong to one of those three superclasses. This answers the question: which graphs are ``almost'' perfect?}, language = {en} }