On kernels in perfect graphs


A kernel of a digraphD is a set of vertices which is both independent and absorbant. In 1983, C. Berge and P. Duchet conjectured that an undirected graphG is perfect if and only if the following condition is fulfilled: ifD is an orientation ofG (where pairs of opposite arcs are allowed) and if every clique ofD has a kernel thenD has a kernel. We prove here the conjecture for the complements of strongly perfect graphs and establish that a minimal counterexample to the conjecture is not a complete join of an independent set with another graph.

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  1. [1]

    C. Berge, V. Chvátal (Eds.):Topics on Perfect Graphs, Ann. of Disc. Math. 21, 1984.

  2. [2]

    C. Berge, P. Duchet: Problème, Séminaire MSH, Paris, Janvier 1983.

  3. [3]

    C. Berge, P. Duchet: Solvability of perfect graphs., Proceedings of the Burnside-Raspail meeting, (Barbados 1986) McGill University, Montréal, 1987.

    Google Scholar 

  4. [4]

    C. Berge, P. Duchet: Kernels and perfect graphs, Bull. Inst. Math. Acad. Sinica 164, 1988, 355–366.

    Google Scholar 

  5. [5]

    M. Blidia: Contribution à l'étude des noyaux dans les graphes, Thesis, Univ. Paris 6, 1984.

  6. [6]

    M. Blidia: Kernels in parity graphs with an orientation condition, submitted.

  7. [7]

    M. Blidia: A parity digraph has a kernel,Combinatorica 6, (1986), 23–27.

    Google Scholar 

  8. [8]

    M. Blidia, P. Duchet, F. Maffray: Meyniel graphs are kernel-M-solvable, submitted.

  9. [9]

    C. Champetier: Kernels in some orientations of comparability graphs,J. of Comb. Theory B 47 (1989), 111–113.

    Google Scholar 

  10. [10]

    P. Duchet: Graphes noyaux-parfaits,Ann. Discr. Math. 9, (1980), 93–101.

    Google Scholar 

  11. [11]

    P. Duchet: Parity graphs are kernel-M-solvable,J. Comb. Theory (B)43, (1980), 121–126.

    Google Scholar 

  12. [12]

    H. Galeana-Sanchez, V. Neuman-Lara: On kernels and semi-kernels of digraphs,Disc. Math. 48, (1984), 67–76.

    Google Scholar 

  13. [13]

    F. Maffray: Sur l'existence des noyaux dans les graphes parfaits, Thesis, Univ. Paris 6 1984.

  14. [14]

    F. Maffray On kernels ini-triangulated graphs,Disc. Math. 61, (1986), 247–251.

    Google Scholar 

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Blidia, M., Duchet, P. & Maffray, F. On kernels in perfect graphs. Combinatorica 13, 231–233 (1993). https://doi.org/10.1007/BF01303206

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