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## The strong perfect graph theorem (2006)

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Venue: | ANNALS OF MATHEMATICS |

Citations: | 279 - 21 self |

### Citations

215 |
Decomposition of regular matroids
- Seymour
- 1980
(Show Context)
Citation Context ... therefore need a fifth basic class) and also we include homogeneous pairs. How can we prove a theorem of the form of 1.3? There are several other theorems of this kind in graph theory — for example, =-=[7, 10, 17, 21, 22, 24, 25]-=- and others. All these theorems say that “every graph (or matroid) not containing an object of type X either falls into one of a few basic classes or admits a decomposition”. And for each of these the... |

129 |
Färbung von graphen, deren samtliche bzw. Deren ungerade kreise starr sind
- Berge
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Citation Context ...n theory (finding the “Shannon capacity” of a graph — it lies between the size of the largest clique and the chromatic number, and so for a perfect graph it equals both). In particular, in 1961 Berge =-=[1]-=- proposed two celebrated conjectures about perfect graphs. Since the second implies the first, they were known as the “weak” and “strong” perfect graph conjectures respectively, although both are now ... |

114 |
Über Graphen und ihre Anwendung auf Determinantentheorie un Mengenlehre, Math.Ann 77
- König
- 1916
(Show Context)
Citation Context ...that if G is a double split graph then so is G.) It is easy to see that all basic graphs are perfect. (For bipartite graphs it is trivial; for line graphs of bipartite graphs it is a theorem of König =-=[15]-=-; for their complements it follows from Lovász’ theorem 1.1, although originally these were separate theorems of König; and for double split graphs we leave it to the reader.) Now we turn to the vario... |

99 |
A Characterization of Perfect Graphs
- Lovász
- 1972
(Show Context)
Citation Context ...es respectively, although both are now theorems, the following: 1.1 The complement of every perfect graph is perfect. 1.2 A graph is perfect if and only if it is Berge. The first was proved by Lovász =-=[16]-=- in 1972. The second, the strong perfect graph conjecture, received a great deal of attention over the past 40 years, but remained open until now, and is the main theorem of this paper. It is easy to ... |

98 |
Star-cutsets and perfect graphs
- Chvátal
- 1985
(Show Context)
Citation Context ...rtices 2sin A with interior in B. A set X ⊆ V (G) is connected if G|X is connected (so ∅ is connected); and anticonnected if G|X is connected. The third kind of decomposition we use is due to Chvátal =-=[6]-=- — a skew partition in G is a partition (A, B) of V (G) such that A is not connected and B is not anticonnected. Despite their elegance, skew partitions pose a difficulty that the other two decomposit... |

98 | Über eine Eigenschaft der ebenen Komplexe - Wagner - 1937 |

71 | Pfaffian orientations, and even directed circuits
- Robertson, Seymour, et al.
(Show Context)
Citation Context ... therefore need a fifth basic class) and also we include homogeneous pairs. How can we prove a theorem of the form of 1.3? There are several other theorems of this kind in graph theory — for example, =-=[7, 10, 17, 21, 22, 24, 25]-=- and others. All these theorems say that “every graph (or matroid) not containing an object of type X either falls into one of a few basic classes or admits a decomposition”. And for each of these the... |

65 |
Über eine Eigenschaft der ebenen
- Wagner
- 1937
(Show Context)
Citation Context ... therefore need a fifth basic class) and also we include homogeneous pairs. How can we prove a theorem of the form of 1.3? There are several other theorems of this kind in graph theory — for example, =-=[7, 10, 17, 21, 22, 24, 25]-=- and others. All these theorems say that “every graph (or matroid) not containing an object of type X either falls into one of a few basic classes or admits a decomposition”. And for each of these the... |

37 |
A new Property of Critical Imperfect Graphs and Some Consequences
- Meyniel
- 1987
(Show Context)
Citation Context ...rtition (A, B) such that some vertex of B is adjacent to all other vertices of B. An even pair means a pair of vertices u, v in a graph such that every path between them has even length. It was known =-=[2, 6, 18]-=- that no minimum imperfect graph admits a star cutset or an even pair, and the earlier versions of 1.3 involved these concepts. For instance, in Reed’s PhD thesis [19], the following conjecture appear... |

35 | Bull-free Berge graphs are perfect - Chvátal, Sbihi - 1987 |

35 |
Composition for perfect graphs
- Cornuéjols, Cunningham
- 1985
(Show Context)
Citation Context ...ne of G, G is a line graph of a bipartite graph, or • one of G, G admits a 2-join, or • G admits a skew partition, or • one of G, G has an even pair. More recently, Conforti, Cornuéjols and Vuˇsković =-=[10]-=- proposed a similar conjecture, with the “even pair” alternative replaced by “one of G, G is bipartite”, although without explicitly listing a proposed set of decompositions. Our result 1.3 is essenti... |

34 | Decomposition of balanced matrices
- Conforti, Cornuéjols, et al.
- 1999
(Show Context)
Citation Context ...kew partition conjecture was still open). A counterexample to all these versions of the conjecture was obtained in the early 1990’s by Irena Rusu. At about the same time, Conforti, Cornuéjols and Rao =-=[9]-=- proved a statement analogous to 1.3 for the class of bipartite graphs in which every induced cycle has length a multiple of four, and their theorem involved 2-joins. Since Cornuéjols and Cunningham [... |

26 |
A semi-strong perfect graph theorem
- Reed
- 1986
(Show Context)
Citation Context ... even length. It was known [2, 6, 18] that no minimum imperfect graph admits a star cutset or an even pair, and the earlier versions of 1.3 involved these concepts. For instance, in Reed’s PhD thesis =-=[19]-=-, the following conjecture appears: 1.6 Conjecture: For every perfect graph G, either one of G, G is a line graph of a bipartite graph, or one of them has a star cutset or an even pair. Reed also stud... |

19 | Tutte's edgecolouring conjecture
- Robertson, Seymour, et al.
- 1997
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Citation Context |

14 |
About skew partitions in minimal imperfect graphs
- Roussel, Rubio
(Show Context)
Citation Context ...) is called a component of A, and a maximal anticonnected subset is called an anticomponent of A.) The lemma following is related to results of [14] that were used by Roussel and Rubio in their proof =-=[23]-=- of 2.1. Indeed, lemma 2.2 of [14] has a similar proof, and one could use that lemma to make this proof a little shorter. 1.5 If G is a minimum imperfect graph, then G admits no balanced skew partitio... |

13 |
Berge trigraphs and their applications
- Chudnovsky
- 2003
(Show Context)
Citation Context ...e the homogeneous pair outcome (in the proof of 13.4), and it is natural to ask if homogeneous pairs are really needed. In fact they can be eliminated; one of us (Chudnovsky) showed in her PhD thesis =-=[3, 4]-=- that the following holds: 1.4 For every Berge graph G, either G is basic, or one of G, G admits a proper 2-join, or G admits a balanced skew partition. But the proof of 1.4 is very long (it consists ... |

13 | Pólya’s permanent problem
- McCuaig
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Citation Context |

11 |
Graphs without Odd Holes, Parachutes or Proper Wheels: A Generalization of Meyniel Graphs, and of Line Graphs of Bipartite Graphs, manuscript
- Conforti, Cornuéjols
(Show Context)
Citation Context ...s is a double diamond, contrary to G ∈ F10. This proves 23.5. Let us mention a theorem of [12], which could be applied at this stage as an alternative to the next section, the following (and see also =-=[8]-=- for some related material): 23.6 Let G ∈ F5. Suppose that for every hole C in G of length ≥ 6, and every vertex v ∈ V (G) \ V (C), either: • v has ≤ 3 neighbours in C, or • v has exactly 4 neighbours... |

10 |
Some properties of minimal imperfect graphs
- Hoàng
- 1996
(Show Context)
Citation Context .... (A maximal connected subset of a nonempty set A ⊆ V (G) is called a component of A, and a maximal anticonnected subset is called an anticomponent of A.) The lemma following is related to results of =-=[14]-=- that were used by Roussel and Rubio in their proof [23] of 2.1. Indeed, lemma 2.2 of [14] has a similar proof, and one could use that lemma to make this proof a little shorter. 1.5 If G is a minimum ... |

9 | Progress on perfect graphs
- Chudnovsky, Robertson, et al.
(Show Context)
Citation Context ...ucture of minimum imperfect graphs. For the latter, linear programming methods have been particularly useful; there are rich connections between perfect graphs and linear and integer programming (see =-=[5, 20]-=- for example). But a third approach has been developing in the perfect graph community over a number of years; the attempt to show that every Berge graph either belongs to some well-understood basic c... |

8 |
A Note on Even Pairs
- Reed
- 1987
(Show Context)
Citation Context ...rtition (A, B) such that some vertex of B is adjacent to all other vertices of B. An even pair means a pair of vertices u, v in a graph such that every path between them has even length. It was known =-=[2, 6, 18]-=- that no minimum imperfect graph admits a star cutset or an even pair, and the earlier versions of 1.3 involved these concepts. For instance, in Reed’s PhD thesis [19], the following conjecture appear... |

6 | Decomposing Berge graphs containing no proper wheels, stretchers or their complements, preprint
- Conforti, Cornuéjols, et al.
- 2002
(Show Context)
Citation Context ...d4, d5} are c1d1, c1d5, c2d2, c2d4, c4d2, c4d4, c5d1, c5d5; and so the subgraph induced on these eight vertices is a double diamond, contrary to G ∈ F10. This proves 23.5. Let us mention a theorem of =-=[12]-=-, which could be applied at this stage as an alternative to the next section, the following (and see also [8] for some related material): 23.6 Let G ∈ F5. Suppose that for every hole C in G of length ... |

5 |
Decomposing Berge graphs containing proper wheels
- Conforti, Cornuéjols, et al.
- 2002
(Show Context)
Citation Context ... set disjoint from C • there are two disjoint Y -complete edges of C. We need to study how the remainder of a recalcitrant graph can attach onto a wheel. (Conforti, Cornuéjols, Vuˇsković and Zambelli =-=[11]-=- also made such a study, and their paper contains results related to ours.) We call C the rim and Y the hub of the wheel. A maximal path in a path or hole H whose vertices are all Y -complete is calle... |

4 | F"arbung von Graphen deren s"amtliche bzw. deren ungerade Kreise starr sind, Wiss - Berge - 1961 |

2 |
Square-free perfect graphs", manuscript
- Conforti, Cornuejols, et al.
- 2001
(Show Context)
Citation Context ...structure of a minimal counterexample. (There are rich connections with linear and integer programming - see [10] for example.) Our approach is different. Recently, Conforti, Cornuéjols and Vuˇsković =-=[5]-=- conjectured that every Berge graph either falls into one of four well-understood classes, or it admits one of several kinds of decomposition. They pointed out that if this could be proved, and if als... |

1 |
Berge trigraphs”, manuscript
- Chudnovsky
- 2004
(Show Context)
Citation Context ...e the homogeneous pair outcome (in the proof of 13.4), and it is natural to ask if homogeneous pairs are really needed. In fact they can be eliminated; one of us (Chudnovsky) showed in her PhD thesis =-=[3, 4]-=- that the following holds: 1.4 For every Berge graph G, either G is basic, or one of G, G admits a proper 2-join, or G admits a balanced skew partition. But the proof of 1.4 is very long (it consists ... |

1 | Berge trigraphs", manuscript - Chudnovsky - 2004 |

1 | P'olya's permanent problem", preprint - McCuaig - 1996 |

1 |
Bull-free Berge graphs are perfect
- unknown authors
- 1985
(Show Context)
Citation Context ...vertices in A with interior in B. A set X ⊆ V (G) is connected if G|X is connected (so ∅ is connected); and anticonnected if G|X is connected. The third kind of decomposition we use is due to Chvátal =-=[2]-=- — a skew partition in G is a partition (A,B) of V (G) so that A is not connected and B is not anticonnected. Skew partitions pose a difficulty that the other two decompositions do not, for it has not... |