Algorithms Finding Tree-Decompositions of Graphs

A graph G has tree-width at most w if it admits a tree-decomposition of width ≤ w. It is known that once we have a tree-decomposition of a graph G of bounded width, many NP -hard problems can be solved for G in linear time. For w ≤ 3 we give a lineartime algorithm for finding such a decomposition and for a general fixed w we obtain a probabilistic algorithm with execution time O(n log n+ n logn| log p|), which for a graph G on n vertices and a real number p > 0 either finds a tree-decomposition of width ≤ 6w or answers that the tree-width of G is ≥ w; this second answer may be wrong but with probability at most p. The second result is based on a separator technique which may be of independent interest.