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Relaxing Perfectness: Which Graphs are 'Almost' Perfect?

Please always quote using this URN: urn:nbn:de:0297-zib-6700
  • 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?

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Author:Annegret Wagler
Document Type:ZIB-Report
Tag:perfect graph; relaxations; stable set polytope
MSC-Classification:05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C17 Perfect graphs
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C10 Integer programming
Date of first Publication:2002/01/22
Series (Serial Number):ZIB-Report (02-03)
Published in:Appeared in: The Sharpest Cut - The Impact of Manfred Padberg and His Work. M. Grötschel (ed.) MPS-SIAM Series on Optimiztion, 2004, pp. 77-96
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