Critical and Anticritical Edges in Perfect Graphs

  • Annegret Wagler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2204)


We call an edge e of a perfect graph G critical if G - e is imperfect and sayfurther that e is anticritical with respect to the complementary graph \( \overline G \) . We ask in which perfect graphs critical and anticritical edges occur and how to find critical and anticritical edges in perfect graphs. Finally, we study whether we can order the edges of certain perfect graphs such that deleting all the edges yields a sequence of perfect graphs ending up with a stable set.


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  1. 1.
    Berge, C.: Färbungen von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wiss. Zeitschrift der Martin-Luther-Universität Halle-Wittenberg (1961) 114–115Google Scholar
  2. 2.
    Berge, C., Duchet, P.: Strongly Perfect Graphs. In: Berge, C., Chvátal, V. (eds.): Topics on Perfect Graphs. North Holland, Amsterdam (1984) 57–61CrossRefGoogle Scholar
  3. 3.
    Chvátal, V.: Star-Cutsets and Perfect Graphs. J. Combin. Theory(B) 39 (1985) 189–199zbMATHCrossRefGoogle Scholar
  4. 4.
    Chvátal, V., Sbihi, N.: Bull-Free Berge Graphs are Perfect. Graphs and Combinatorics 3 (1987) 127–139zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Grötschel, M., Lovász, L., Schrijver, A.: The Ellipsoid Method and its Consequences in Combinatorial Optimization. Combinatorica 1 (1981) 169–197zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hayward, R.B.: Weakly Triangulated Graphs. J. Combin. Theory(B) 39 (1985) 200–209zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hayward, R.B.: Generating Weakly Triangulated Graphs. J. Graph Theory 21, No. 1 (1996) 67–69Google Scholar
  8. 8.
    Hayward, R.B., Hoàng, C.T., Maffray, F.: Optimizing Weakly Triangulated Graphs. Graphs and Combinatorics 5 (1989) 339–349, erratum in 6 (1990) 33–35Google Scholar
  9. 9.
    Hertz, A.: Slim Graphs. Graphs and Combinatorics 5 (1989) 149–157zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hoàng, C.T.: Some Properties of Minimal Imperfect Graphs. Discrete Math. 160 (1996) 165–175zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lovász, L.: Normal Hypergraphs and the Weak Perfect Graph Conjecture. Discrete Math. 2 (1972) 253–267zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Meyniel, H.: On The Perfect Graph Conjecture. Discrete Math. 16 (1976) 339–342CrossRefMathSciNetGoogle Scholar
  13. 13.
    Meyniel, H.: A New Property of Critical Imperfect Graphs and some Consequences. Europ. J. Combinatorics 8 (1987) 313–316zbMATHMathSciNetGoogle Scholar
  14. 14.
    Olariu, S.: No Antitwins in Minimal Imperfect Graphs. J. Combin. Theory(B) 45 (1988) 255–257zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Padberg, M.W.: Perfect Zero-One Matrices. Math. Programming 6 (1974) 180–196zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Parthasarathy, K.R., Ravindra, G.: The Strong Perfect Graph Conjecture is True for K1,3-free Graphs. J. Combin. Theory(B) 21 (1976) 212–223zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Spinrad, J., Sritharan, R.: Algorithms for WeaklyT riangulated Graphs. Discrete Appl. Math. 59 (1995) 181–191zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Tucker, A.: Critical Perfect Graphs and Perfect 3-chromatic Graphs. J. Combin. Theory(B) 23 (1977) 143–149zbMATHCrossRefGoogle Scholar
  19. 19.
    Tucker, A.: Coloring Perfect (K4-e)-free Graphs. J. Combin. Theory(B) 42 (1987) 313–318zbMATHCrossRefGoogle Scholar
  20. 20.
    Wagler, A.: Critical Edges in Perfect Graphs. PhD thesis, TU Berlin (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Annegret Wagler
    • 1
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik BerlinBerlinGermany

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