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The strong perfect graph conjecture
 Proceedings of the ICM
, 2002
"... A graph is perfect if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called basic. In 1 ..."
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. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak perfect graph conjecture, which states that a graph is perfect if and only if its complement is perfect, was proved in 1972 by Lovász. This result is now known as the perfect graph theorem. The strong
Forcing Colorations and the Strong Perfect Graph Conjecture
"... We give various reformulations of the Strong Perfect Graph Conjecture, based on a study of forced coloring procedures, uniquely colorable subgraphs and ! \Gamma 1cliques in minimal imperfect graphs. ..."
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We give various reformulations of the Strong Perfect Graph Conjecture, based on a study of forced coloring procedures, uniquely colorable subgraphs and ! \Gamma 1cliques in minimal imperfect graphs.
How the proof of the strong perfect graph conjecture was found
, 2006
"... In 1961, Claude Berge proposed the “strong perfect graph conjecture”, probably the most beautiful open question in graph theory. It was answered just before his death in 2002. This is an overview of the solution, together with an account of some of the ideas that eventually brought us to the answer. ..."
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In 1961, Claude Berge proposed the “strong perfect graph conjecture”, probably the most beautiful open question in graph theory. It was answered just before his death in 2002. This is an overview of the solution, together with an account of some of the ideas that eventually brought us to the answer.
The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution
, 2010
"... The Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The rst of these three approaches ..."
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The Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The rst of these three
About the Strong Perfect Graph Conjecture on circular partitionable graphs
, 1999
"... There is a construction due to Chv'atal, Graham, Perold and Whitesides for making partitionable graphs with circular symetry. It is known that this construction doesn't produce any counterexample to Berge's conjecture. Grinstead conjectured that any circular partitionable graph is g ..."
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There is a construction due to Chv'atal, Graham, Perold and Whitesides for making partitionable graphs with circular symetry. It is known that this construction doesn't produce any counterexample to Berge's conjecture. Grinstead conjectured that any circular partitionable graph
CLASSES OF HYPERGRAPHS, THE STRONG PERFECT GRAPH CONJECTURE AND RELATIONAL DATABASES
, 1986
"... ..."
The strong perfect graph theorem
 ANNALS OF MATHEMATICS
, 2006
"... A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asse ..."
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Cited by 287 (21 self)
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A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961
ALMOST PERFECT MATRICES AND GRAPHS
, 1998
"... We introduce the notions of!projection andprojection that map almost integral polytopes associated with almost perfect graphs G with n nodes from R n into R n,! where! is the maximum clique size in G. We show that C. Berge's strong perfect graph conjecture is correct if and only if the projec ..."
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We introduce the notions of!projection andprojection that map almost integral polytopes associated with almost perfect graphs G with n nodes from R n into R n,! where! is the maximum clique size in G. We show that C. Berge's strong perfect graph conjecture is correct if and only
Chairfree Berge Graphs are Perfect
 GRAPHS COMBIN
, 1996
"... 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 ..."
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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
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices
Results 1  10
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