Canonical tree-decompositions of finite graphs I. Existence and algorithms

Abstract

We construct tree-decompositions of graphs that distinguish all their k-blocks and tangles of order k, for any fixed integer k. We describe a family of algorithms to construct such decompositions, seeking to maximize their diversity subject to the requirement that they commute with graph isomorphisms. In particular, all the decompositions constructed are invariant under the automorphisms of the graph.

Quasi-4-connected components

“…It was shown in [17,1] that for every k, every graph admits a canonical decomposition into its tangles of order k. Related to this is the decomposition into so-called (k − 1)blocks due to [3]. These decompositions (for k = 4) are related to ours.…”

“…In [5] I exploited some of the main ideas underlying our Decomposition Theorem to obtain such simplifications in the context of logical definability, and I believe the Decomposition Theorem proved here may turn out to be similarly useful in an algorithmic context. 1…”

A Short Derivation of the Structure Theorem for Graphs with Excluded Topological Minors

“…A k-block in a graph G is a set B of at least k vertices which is inclusion-maximal with the property that for every separation (U, W ) of order < k, we either have B ⊆ U or B ⊆ W . The notion of a k-block, which was first studied by Mader [14,13], has previously been considered in the study of graph decompositions [4,3,5].…”

“…Therefore, if we assume our graph does not contain a subdivision of K r , then we can separate any large minor from every large block. It then follows from the tangle tree theorem of Robertson and Seymour [15] -or rather its extension to profiles [11,7,3] -that there exists a tree-decomposition which separates the blocks from the minors. Hence each part is either free of large minors or of large blocks.…”

Tangles and Connectivity in Graphs

“…This does not sound particularly interesting, but the grid could be a subgraph of a larger graph, and then the tangle would identify it as a highly connected region within that graph. A key theorem about tangles is that every graph admits a canonical tree decomposition into its tangles of order k [1,21]. This can be seen as a generalisation of the decomposition of a graph into its 3-connected components.…”