Before revision: We give an algorithm, which given an n-vertex graph G and an integer k, in time 2^{O(k)} n either outputs a tree decomposition of G of width at most 2k + 1 or determines that the treewidth of G is larger than k  ]  .
After revision: We give an algorithm, which given an n-vertex graph G and an integer k, in time 2^{O(k)} n either outputs a tree decomposition of G of width at most 2k + 1 or determines that the treewidth of G is larger than k ]. Our algorithm is based on a local improvement method adapted from a proof of Bellenbaum and Diestel .