Some properties of minimal imperfect graphs
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The Odd Pair Conjecture, with ‘even’ replaced by ‘odd’, is the natural analogue of the Even Pair Lemma and is proved a partial result for this conjecture, namely: no minimal imperfect graph G contains a three-pair, i.e. two nonadjacent vertices u 1 , u 2 such that all chordless paths of G joining u 1 to u 2 contain precisely three edges.About:
This article is published in Discrete Mathematics.The article was published on 1996-11-15 and is currently open access. It has received 26 citations till now. The article focuses on the topics: Perfect graph theorem & Perfect graph.read more
Citations
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The Strong Perfect Graph Theorem
TL;DR: The strong perfect graph conjecture as discussed by the authors states that 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 sub graph of G is an odd cycle of length at least five or the complement of one.
Posted Content
The strong perfect graph theorem
TL;DR: In this article, the authors prove that every Berge graph either falls into one of a few basic classes, or it has a kind of separation that cannot occur in a minimal imperfect graph.
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Progress on perfect graphs
TL;DR: The main aspects of perfect graphs and their relevance are surveyed and the recent proof of the Strong Perfect Graph Conjecture of Berge from 1961 is outlined.
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Finding Skew Partitions Efficiently
TL;DR: This work presents a polynomial-time algorithm for testing whether a graph admits a skew partition and solves the more general list skew partition problem, where the input contains a list containing some of the labels A,B,C,D of the four parts.
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Square-free perfect graphs
TL;DR: It is proved that square-free perfect graphs are bipartite graphs or line graphs of bipartITE graphs or have a 2-join or a star cutset.
References
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Book
Algorithmic graph theory and perfect graphs
TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.
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Normal hypergraphs and the perfect graph conjecture
TL;DR: In this paper, it was shown that the complement of a perfect graph is perfect and a new proof for a related theorem of Berge and Las Vergnas was given for integer valued linear programming.
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Perfect zero---one matrices
TL;DR: A characterization of perfect zero–one matrices in terms offorbidden submatrices is given, closely related to perfect graphs and constitute a generalization of balanced matrices as introduced by C. Berge.
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Star-cutsets and perfect graphs
TL;DR: This work first establishes a certain property of minimal imperfect graphs and then uses it to generate large classes of perfect graphs, which are then used to establish and generate large numbers of minimal perfect graphs.
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Anti-blocking polyhedra
TL;DR: In this paper, a theory parallel to that for blocking pairs of polyhedra is developed for anti-blocking pairs, and certain combinatorial results and problems are discussed in this framework.