Two Short Proofs Concerning Tree-Decompositions

The main purpose of this note is to give a short proof of Thomas’s theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width [ 9 ]. This is a useful tool in the theory of tree-decompositions; for example, it is a key lemma in Robertson & Seymour’s proof of the fact that every set of graphs of bounded tree-width is well-quasi-ordered [ 7 ]. This latter result is the starting point for the proof of their graph minor theorem, see [ 2 ]. It is also the first step in the now available short proof of the ‘general Kuratowski theorem’ that embeddability in any fixed surface is characterized by finitely many forbidden minors (combine it with [ 3 ] and either [ 5 ] or [ 10 ]), a main corollary of the graph minor theorem. Another (more constructive) short proof of Thomas’s theorem has been given in [ 1 ]. An analogous result for branch-width was obtained by Geelen, Gerards & Whittle [ 4 ], also with a short and simple proof. The ‘branch-width’ of a graph is a parameter closely related to tree-width but not 1–1 translatable, so the result proved in [ 4 ] does not imply Thomas’s theorem as reproved in this paper. But [ 4 ] does give a complete short proof (including the WQO part) of the above-mentioned result from [ 7 ], where exact bounds for Thomas’s theorem are not required. Our proof of Thomas’s theorem differs from the original in that we use a simpler induction parameter. This simplifies the induction step: we have less to verify, and the presentation becomes considerably less technical. The tree-decomposition we use, however, is the same as in [ 9 ]. Essentially, it is constructed recursively from tree-decompositions of two subgraphs by adding their intersection as a new part to both decompositions, to serve as the common interface required for their amalgamation. Whether or not this can be done without increasing the width is a key issue in the study of tree-decompositions more generally. Thomas’s sufficient condition under which this technique can be applied is a contribution of independent use and interest, and so we have extracted it into a separate lemma (Lemma 2). This lemma (with some inessential additional details) has been used again before, in Seymour & Thomas’s proof of what Reed [ 6 ] has called their treewidth duality theorem [ 8 ]. A streamlined proof of this result has already ap-