Abstract
A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Chvátal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs. We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile. Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphs
Similar content being viewed by others
References
Berge, C.: Les problèmes de coloration en théorie des graphes. Publ. Inst. Stat. Univ. Paris9, 123–160 (1960)
Bertschi, M.E.: Perfectly contractile graphs. J. Comb. Theory Ser.B 50, 222–230 (1990)
Chvátal, V.: Perfectly ordered graphs. In Berge C. and V. Chvátal, editors, Topics on Perfect Graphs, pages 63–65. North-Holland. Amsterdam, 1984
Chvátal, V.: A class of perfectly orderable graphs. Technical Report 89573-OR, Forschungsbereich für Diskrete Mathematik, Institut für Ökonometrie und Operations Research, Rheinische Friedrich Wilhelms Universitaet, Bonn, Germany, May 1989
Chvátal, V., Sbihi, N.: Bull-free Berge graphs are perfect. Graph. Comb.3, 127–139 (1987)
Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput.14, 926–934 (1985)
De Figueiredo, C.M.H.: Um Estudo de Problemas Combinatórios em Grafos Perfeitos. PhD thesis, COPPE/UFRJ, Rio de Janeiro, 1991. In Portuguese
Fonlupt, J., Uhry, J.P.: Transformations which preserve perfectness andh-perfectness of graphs. Annals of Disc. Math.16, 83–85 (1982)
Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. In C. Berge and V. Chvátal, editors, Topics on Perfect Graphs (Ann. Discrete Math. 21), pages 325–356. North-Holland, Amsterdam, 1984
Hayward, R.: Weakly triangulated graphs. J. Comb. Theory Ser.B 39, 200–209 (1985)
Hayward, R., Hoàng, C.T., Maffray, F.: Optimizing weakly triangulated graphs. Graph. Comb.5, 339–349 (1989). See erratum in vol. 6, 1990, p. 33–35
Hertz, A., de Werra, D.: Perfectly orderable graphs are quasi-parity graphs: a short proof. Disc. Math.68, 111–113 (1988)
Hoàng, C.T.: Algorithms for minimum weighted coloring of perfectly ordered, comparability, triangulated and clique-separable graphs. Technical Report 90832, Forschungsinstitut für Diskrete Mathematik, Institut für Operations Research, Universitaet Bonn, Germany, March 1990. Disc. Appl. Math. (to be published)
Meyniel, H.: A new property of critical imperfect graphs and some consequences. Europ. J. Comb.8, 313–316 (1987)
Reed, B.A.: Problem session on parity problems. Perfect Graphs Workshop, Princeton University, New Jersey, June 1993
Reed, B.A., Sbihi, N.: Recognizing bull-free perfect graphs. preprint, 1990
Seinsche, S.: On a property of the class ofn-colorable graphs. J. Comb. Theory Ser. B16, 191–193 (1974)
Spinrad, J.:P 4-trees and substitution decomposition. Disc. Appl. Math.39, 263–291 (1992)
Author information
Authors and Affiliations
Additional information
Partially supported by CNPq, grant 30 1160/91.0
Rights and permissions
About this article
Cite this article
de Figueiredo, C.M.H., Maffray, F. & Porto, O. On the structure of bull-free perfect graphs. Graphs and Combinatorics 13, 31–55 (1997). https://doi.org/10.1007/BF01202235
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01202235