Mathematical Programming

, Volume 171, Issue 1–2, pp 519–522 | Cite as

Polyhedral results on the stable set problem in graphs containing even or odd pairs

  • Jonas T. Witt
  • Marco E. Lübbecke
  • Bruce Reed
Short Communication Series A


Even and odd pairs are important tools in the study of perfect graphs and were instrumental in the proof of the Strong Perfect Graph Theorem. We suggest that such pairs impose a lot of structure also in arbitrary, not just perfect graphs. To this end, we show that the presence of even or odd pairs in graphs imply a special structure of the stable set polytope. In fact, we give a polyhedral characterization of even and odd pairs.


Stable set problem Stable set polytope Even pairs Odd pairs Integer programming 

Mathematics Subject Classification

90C10 (Integer programming) 90C27 (Combinatorial optimization) 90C57 (Polyhedral combinatorics, branch-and-bound, branch-and-cut) 


  1. 1.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006). doi: 10.4007/annals.2006.164.51 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chudnovsky, M., Seymour, P.: Even pairs in Berge graphs. J. Combin. Theory Ser. B 99(2), 370–377 (2009). doi: 10.1016/j.jctb.2008.08.002 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chvátal, V.: On certain polytopes associated with graphs. J. Combin. Theory Ser. B 18(2), 138–154 (1975). doi: 10.1016/0095-8956(75)90041-6 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fulkerson, D.R.: Blocking and anti-blocking pairs of polyhedra. Math. Program. 1(1), 168–194 (1971). doi: 10.1007/BF01584085 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discrete Math. 306(10–11), 867–875 (2006). doi: 10.1016/j.disc.2006.03.007 CrossRefzbMATHGoogle Scholar
  6. 6.
    Meyniel, H.: A new property of critical imperfect graphs and some consequences. Eur. J. Combin. 8(3), 313–316 (1987). doi: 10.1016/S0195-6698(87)80037-9 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Meyniel, H., Olariu, S.: A new conjecture about minimal imperfect graphs. J. Combin. Theory Ser. B 47(2), 244–247 (1989). doi: 10.1016/0095-8956(89)90024-5 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ramirez-Alfonsin, J.L., Reed, B.A. (eds.): Perfect Graphs. Wiley, Chichester, UK (2001)Google Scholar
  9. 9.
    Reed, B.: Perfection, parity, planarity, and packing paths. In: Proceedings of the 1st Integer Programming and Combinatorial Optimization Conference, pp. 407–419. University of Waterloo Press (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  • Jonas T. Witt
    • 1
  • Marco E. Lübbecke
    • 1
  • Bruce Reed
    • 2
    • 3
  1. 1.Operations ResearchRWTH Aachen UniversityAachenGermany
  2. 2.Laboratoire I3S CNRSSophia-AntipolisFrance
  3. 3.IMPARio de JaneiroBrazil

Personalised recommendations