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ISSN 1088-9485(e) ISSN 0273-0979(p)

Graph minor theory

Author(s): László Lovász
Journal: Bull. Amer. Math. Soc. 43 (2006), 75-86.
MSC (2000): Primary 05C83
Posted: October 24, 2005
MathSciNet review: 2188176
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory.

The motivating problem was Kuratowski's characterization of planar graphs, and a far-reaching generalization of this, conjectured by Wagner: If a class of graphs is minor-closed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minor-closed class of graphs does not contain all graphs, then every graph in it is glued together in a tree-like fashion from graphs that can almost be embedded in a fixed surface.

We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.

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