Related papers: Canonical graph decompositions via local separatio…
We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as…
We prove that every graph has a canonical tree of tree-decompositions that distinguishes all principal tangles (these include the ends and various kinds of large finite dense structures) efficiently. Here `trees of tree-decompositions' are…
While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits…
We show that a locally finite, connected graph $G$ is $r$-locally chordal (that is, its $r/2$-balls are chordal) if and only if the unique canonical graph-decomposition $\mathcal{H}_r(G)$ of $G$ displaying its $r$-global structure is into…
A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by removing less than $k$ vertices. It is separable if there exists a tree-decomposition of adhesion less than $k$ of $G$ in which…
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…
Robertson and Seymour constructed for every graph $G$ a tree-decomposition that efficiently distinguishes all the tangles in $G$. While all previous constructions of these decompositions are either iterative in nature or not canonical, we…
A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph $G$ is a decomposition $\mathcal{D}$ of $G$ such that every subgraph $H \in \mathcal{D}$ is locally irregular.…
We show that every graph admits a canonical tree-like decomposition into its $k$-edge-connected pieces for all $k\in\mathbb{N}\cup\{\infty\}$ simultaneously.
In Part I of this series we described three algorithms that construct canonical tree-decompositions of graphs which distinguish all their k-blocks and tangles of order k. We now establish bounds on the number of parts in these…
We offer a new structural basis for the theory of 3-connected graphs, providing a unique decomposition of every such graph into parts that are either quasi 4-connected, wheels, or thickened $K_{3,m}$'s. Our construction is explicit,…
A graph $G$ is said to be ubiquitous, if every graph $\Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every…
We provide a unique decomposition of every 4-connected graph into parts that are either quasi-5-connected, cycles of triangle-torsos and 3-connected torsos on $\leq 5$ vertices, generalised double-wheels, or thickened $K_{4,m}$'s. The…
A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e…
We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with…
The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the…
The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever…
We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem…
We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Bar\'at and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a…
We present a systematic investigation into how tree-decompositions of finite adhesion capture topological properties of the space formed by a graph together with its ends. As main results, we characterise when the ends of a graph can be…