A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth

Reviewer: S. Srinivasan

A well <__?__Pub Caret>known problem in graph theory is, Given a graph G and an integer k , is the treewidth of G at most k __?__ This problem was shown to be NP-complete in 1987. Since then, many researchers have attempted to find a linear-time algorithm for special values of k . In this paper, the author shows that for a constant k , a linear-time algorithm exists that checks to see if a given graph G has a treewidth of at most k . If so, it finds such a tree decomposition. An important corollary of this result is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. The main result is proven in a series of lemmas. Along the way, several definitions are introduced and explained. This paper relies heavily on the work of Robertson and Seymour. The proofs are easy to follow. An extensive list of references helps in following the work in this area. The paper will be of interest to many graph theorists.