@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} } @misc{Wagler2002, author = {Wagler, Annegret}, title = {Antiwebs are Rank-Perfect}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6742}, number = {02-07}, year = {2002}, abstract = {\We discuss a nested collection of three superclasses of perfect graphs: near-perfect, rank-perfect, and weakly rank-perfect graphs. For that, we start with the description of the stable set polytope for perfect graphs and allow stepwise more general facets for the stable set polytopes of the graphs in each superclass. Membership in those three classes indicates how far a graph is away from being perfect. We investigate for webs and antiwebs to which of the three classes they belong. We provide a complete description of the facets of the stable set polytope for antiwebs (with help of a result due to Shepherd on near-bipartite graphs). The main result is that antiwebs are rankperfect.}, language = {en} }