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Chair-Free Berge Graphs Are Perfect

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Abstract

A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge’s Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd}, eb. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect Graph Conjecture is true for chair-free graphs.

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References

  1. Berge, C.: Färbung von Graphen, deren sämlitche bzw, ungerade Kreise starr sind (Zusammenfassung), Wiss, Z. Mar-tin Luther Univ. Halle Wittenberg, Math. Nat. Reihe (1961) 114.

  2. Berge, C., Chvátal, V. Eds.: Topics on Perfect Graphs. Annals of Discrete Mathematics, Vol 21 Amsterdam: North Holland 1984

  3. Conforti, M., Cornuéjols, G., Kapoor, A., Vušković, K.: A Mickey Mouse Decomposition Theorem, Proceedings IPCO IV, Copenhagen, (May 1995)

  4. Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory ser. B 18, 138–154 (1975)

    Article  Google Scholar 

  5. Chvátal, V.: Star cutsets and perfect graphs. J. Comb. Theory ser. B 39, 189–299 (1985)

    Article  Google Scholar 

  6. Chvátal, V., Sbihi, N.: Bull-free Berge graphs are perfect. Graphs and Comb. 3, 127–139 (1987)

    Article  MATH  Google Scholar 

  7. De Simone, C., Galluccio, A.: New classes of Berge perfect graphs. Discrete Math. 131, 67–79 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. De Simone, C., Sassano, A.: Stability Number of Bull and Chair Free Graphs. Discrete Appl. Math. 41, 121–129 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Galluccio, A., Sassano, A.: The rank facets of the stable set polytope for claw-free graphs. Technical Report n. 340, IASI-CNR, Rome (1994) J. Comb. Theory ser. B (1997) (to be published)

  10. Lovász, L.: Normal Hypergraphs and the perfect graph conjecture. Discrete Math. 2, 253–267 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lovász, L.: A Characterization of Perfect Graphs, J. Comb. Theory ser. B 13, 95–98 (1972)

    Article  Google Scholar 

  12. Lovász, L., Plummer, M.: Matching Theory. Annals of Discrete Mathematics, Vol 29 Amsterdam: North Holland 1986

    Google Scholar 

  13. Mannino, C., Sassano, A.: Edge Projection and the Maximum Cardinality Stable Set Problem. In: D.S. Johnson, M.A. Trick: Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge. DIMACS Series in Discrete Mathematics and Theoretical Computer Science (to be published)

  14. Olariu, S.: On the strong perfect graph conjecture. J. Graph Theory 12, 169–176 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Olariu, S.: Paw-free graphs. Information Processing Letters 28, 53–54 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Olariu, S.: The Strong Perfect Graph Conjecture for Pan-free graphs, J. Comb. Theory ser. B 47, 187–191 (1989)

    Article  MathSciNet  Google Scholar 

  17. Padberg, M.W.: Perfect zero-one Matrices. Mathematical Programming, 6, 180–196 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  18. Parthasarathy, K.R., Ravindra, G.: The strong perfect graph conjecture is true for (K 1, 3)-free graphs. J. Comb. Theory ser. B 21 212–223 (1976)

    Article  MathSciNet  Google Scholar 

  19. Parthasarathy, K.R., Ravindra, G.: The validity of the strong perfect graph conjecture for (K 4 − e)-free graphs. J. Comb. Theory ser. B 26, 98–100 (1979)

    Article  MathSciNet  Google Scholar 

  20. Sassano, A.: Reducible cliques and the Strong Perfect Graph Conjecture, IASI Technical Report R.257, (March 1989)

  21. Sebő, A.: On critical edges in minimal imperfect graphs. Proceedings IPCO III, Erice (May 1993).

  22. Seinsche, D.: On a property of the class of n-colorable graphs. J. Comb. Theory ser. B 16, 191–193 (1974)

    Article  MathSciNet  Google Scholar 

  23. Sun, Liping: Two classes of perfect graphs. J. Comb. Theory ser. B 53, 273–292 (1991)

    Article  Google Scholar 

  24. Tucker, A.: Coloring perfect (K 4 − e)-free graphs. J. Comb. Theory ser. B 42, 313–318 (1987).

    Article  Google Scholar 

  25. Tucker, A.: Critical perfect graphs and perfect 3-chromatic graphs. J. Comb. Theory ser. B 23, 143–149 (1977)

    Article  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, via Buonarroti, 12 - 00185, Roma, Italy

    Antonio Sassano

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  1. Antonio Sassano
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Correspondence to Antonio Sassano.

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This work was partially supported by MURST, Roma, Italy and Progetto Finalizzato Trasporti II, CNR, Italy

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Sassano, A. Chair-Free Berge Graphs Are Perfect. Graphs and Combinatorics 13, 369–395 (1997). https://doi.org/10.1007/BF03353015

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