Related papers: Canonical tree-decompositions of finite graphs I. …
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…
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 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…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested…
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 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…
Roberston and Seymour introduced tangles of order $k$ as objects representing highly connected parts of a graph and showed that every graph admits a tree-decomposition of adhesion $<k$ in which each tangle of order $k$ is contained in a…
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.
Every finite graph $G$ can be decomposed in a canonical way that displays its local connectivity-structure [DJKK26]. These decompositions are defined via a suitable more tree-like covering of $G$, whose tangle-tree structure is projected…
In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most $k$, when an effective…
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…
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…
A tangle of order $k$ in a matroid or graph may be thought of as a "$k$-connected component". For a tangle of order $k$ in a matroid or graph that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or…
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 give a short, topological proof that all graphs admit tree-decompositions displaying their topological ends.
Graph decompositions are the natural generalisation of tree decompositions where the decomposition tree is replaced by a genuine graph. Recently they found theoretical applications in the theory of sparsity, topological graph theory,…
We generalise structure tree theory, which is based on removing finitely many edges, to removing finitely many vertices. This gives a significant generalization of Tutte's tree decomposition of 2-connected graphs into 3-connected blocks.…
In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…