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The perfection and recognition of bull-reducible Berge graphs

Published online by Cambridge University Press:  15 March 2005

Hazel Everett ,
Celina M.H. de Figueiredo ,
Sulamita Klein  and
Bruce Reed
Hazel Everett
Affiliation:
LORIA, France; Hazel.Everett@loria.fr
Celina M.H. de Figueiredo
Affiliation:
Universidade Federal do Rio de Janeiro, Brasil; celina@cos.ufrj.br, sula@cos.ufrj.br
Sulamita Klein
Affiliation:
Universidade Federal do Rio de Janeiro, Brasil; celina@cos.ufrj.br, sula@cos.ufrj.br
Bruce Reed
Affiliation:
McGill University, Canada; breed@cs.mcgill.ca
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Abstract

The recently announced Strong Perfect Graph Theorem states that the class of perfect graphs coincides with the class of graphs containing no induced odd cycle of length at least 5 or the complement of such a cycle. A graph in this second class is called Berge. A bull is a graph with five vertices x, a, b, c, d and five edges xa, xb, ab, ad, bc. A graph is bull-reducible if no vertex is in two bulls. In this paper we give a simple proof that every bull-reducible Berge graph is perfect. Although this result follows directly from the Strong Perfect Graph Theorem, our proof leads to a recognition algorithm for this new class of perfect graphs whose complexity, O(n6), is much lower than that announced for perfect graphs.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 1: Imre Simon, the tropical computer scientist , January 2005 , pp. 145 - 160
Copyright
© EDP Sciences, 2005

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References

C. Berge and V. Chvátal, Topics on Perfect Graphs. North-Holland, Amsterdam, Ann. Discrete Math. 21 (1984).
Chvátal, V., Star-cutsets and perfect graphs. J. Combin. Theory Ser. B 39 (1985) 189–199. CrossRef
M. Chudnovsky, G. Cornuejols, X. Liu, P. Seymour and K. Vuskovic, Cleaning for Bergeness, manuscript (2003).
M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The Strong Perfect Graph Theorem, manuscript (2003).
M. Chudnovsky and P. Seymour, Recognizing Berge graphs, manuscript (2003).
Chvátal, V. and Sbihi, N., Bull-free Berge graphs are perfect. Graphs Combin. 3 (1987) 127–139. CrossRef
de Figueiredo, C.M.H., Maffray, F. and Porto, O., On the structure of bull-free perfect graphs. Graphs Combin. 13 (1997) 31–55. CrossRef
de Figueiredo, C.M.H., Maffray, F. and Porto, O., On the structure of bull-free perfect graphs, 2: The weakly chordal case. Graphs Combin. 17 (2001) 435–456. CrossRef
M. Grötschel, L. Lovász and A. Schrijver, Polynomial algorithms for perfect graphs, in Topics on Perfect Graphs, edited by C. Berge and V. Chvátal. North-Holland, Amsterdam, Ann. Discrete Math. 21 (1984) 325–356.
Hayward, R.B., Discs in unbreakable graphs. Graphs Combin. 11 (1995) 249–254. CrossRef
Hayward, R.B., Bull-free weakly chordal perfectly orderable graphs. Graphs Combin. 17 (2001) 479–500. CrossRef
Jamison, B. and Olariu, S., P 4-reducible graphs – a class of uniquely tree-representable graphs. Stud. Appl. Math. 81 (1989) 79–87. CrossRef
X. Liu, G. Cornuejols and K. Vuskovic, A polynomial algorithm for recognizing perfect graphs, manuscript (2003).
Lovász, L., Normal hypergraphs and the weak perfect graph conjecture. Discrete Math. 2 (1972) 253–267. CrossRef
J.L. Ramirez-Alfonsin and B.A. Reed, Perfect Graphs. Wiley (2001).
Reed, B. and Sbihi, N., Recognizing bull-free perfect graphs. Graphs Combin. 11 (1995) 171–178. CrossRef