Chapter 8 Algorithmic implications of the graph minor theorem
Daniel A. Bienstock, Michael Allen Langston · Handbooks in operations research and management science · 1995
This chapter describes some of the algorithmic ramifications of graph minor theorem and its consequences. Significantly, many of the tools used in the proof of the graph minor theorem can be applied to a very broad class of algorithmic problems. For example, Robertson and Seymour have obtained a relatively simple polynomial-time algorithm for the disjoint paths problem, a task that had eluded researchers for many years. Other applications include combinatorial problems from several domains, including network routing, utilization and design. Indeed, it is a critical measure of the value of the Graph Minor Theorem that so many applications are already known for it. Three types of the key notions employed in theorem are minors, obstructions and well-quasiorders. Kuratowski's theorem may be regarded as a characterization of planarity by means of excluded graphs, henceforth termed obstructions. Characterizations of this nature abound in combinatorial mathematics and optimization. Some familiar examples include the max-flow min-cut theorem, Seymour's description of the clutters with the max-flow min-cut property and Farkas' lemma.