Related papers: A Menger-like property of tree-cut width
This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the…
As a major step in their proof of Wagner's conjecture, Robertson and Seymour showed that every graph not containing a fixed graph $H$ as a minor has a tree-decomposition in which each torso is almost embeddable in a surface of bounded…
Diestel and M\"uller showed that the connected tree-width of a graph $G$, i.e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of $G$ and the largest length of a geodesic cycle in…
Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large…
Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…
When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded ``tree-length''. We will show that this is…
A $T$-decomposition of a graph $G$ is a set of edge-disjoint copies of $T$ in $G$ that cover the edge set of $G$. Graham and H\"aggkvist (1989) conjectured that any $2\ell$-regular graph $G$ admits a $T$-decomposition if $T$ is a tree with…
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 twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin…
We introduce the graph theoretical parameter of edge treewidth. This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of…
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.
It is known that there is a linear dependence between the treewidth of a graph and its balanced separator number: the smallest integer $k$ such that for every weighing of the vertices, the graph admits a balanced separator of size at most…
Many of the tools developed for the theory of tree-decompositions of graphs do not work for directed graphs. In this paper we show that some of the most basic tools do work in the case where the model digraph is a directed path. Using these…
The 2-Decomposition Conjecture, equivalent to the 3-Decomposition Conjecture stated in 2011 by Hoffmann-Ostenhof, claims that every connected graph $G$ with vertices of degree 2 and 3, for which $G \setminus E(C)$ is disconnected for every…
In 2006, Bar\'at and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition…
A graph is chordal if it contains no induced cycle of length four or more. While finite chordal graphs are precisely those admitting tree-decompositions into cliques, this fails for infinite graphs. We establish two results extending the…
Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider…
The Bar\'at-Thomassen conjecture asserts that for every tree $T$ on $m$ edges, there exists a constant $k_T$ such that every $k_T$-edge-connected graph with size divisible by $m$ can be edge-decomposed into copies of $T$. So far this…
Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an…
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph $G=(V,E)$, a multiset…