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 a slightly weaker notion than `tree-decompositions' but much more well-behaved than `tree-like metric spaces'. This theorem is best possible in the sense that we give an example that `trees of tree-decompositions' cannot be strengthened to `tree-decompositions' in the above theorem.
This implies results of Dunwoody and Krön as well as of Carmesin, Diestel, Hundertmark and Stein. Beyond that for locally finite graphs our result gives for each $k\in\mathbb N$ canonical tree-decompositions that distinguish all $k$-distinguishable ends efficiently.
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 decompositions that contain no such block or tangle, and determine conditions under which such parts contain nothing but a k-block.
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.
In matching theory, one of the most fundamental and classical branches of combinatorics, {\em canonical decompositions} of graphs are powerful and versatile tools that form the basis of this theory. However, the abilities of the known canonical decompositions, that is, the {\em Dulmage-Mendelsohn}, {\em Kotzig-Lovász}, and {\em Gallai-Edmonds} decompositions, are limited because they are only applicable to particular classes of graphs, such as bipartite graphs, or they are too sparse to provide sufficient information. To overcome these limitations, we introduce a new canonical decomposition th…
We provide a canonical decomposition for a class of bipartite grafts known as combs. As every bipartite graft is a recursive combination of combs, our results provides a canonical decomposition for general bipartite grafts. Our new decomposition is by definition a generalization of the classical canonical decomposition in matching theory, that is, the Dulmage-Mendelsohn decomposition for bipartite graphs with perfect matchings. However, it exhibits much more complicated structure than its classical counterpart. It is revealed from our results that bipartite grafts has a canonical structure tha…
Let $T$ be a tree, we show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, and use it in order to get formulas for independence number and matching number of a tree. We also prove that the number of maximum matchings in a tree is related to the null decomposition.
In 2019, Dvořák asked whether every connected graph $G$ has a tree decomposition $(T, \mathcal{B})$ so that $T$ is a subgraph of $G$ and the width of $(T, \mathcal{B})$ is bounded by a function of the treewidth of $G$. We prove that this is false, even when $G$ has treewidth $2$ and $T$ is allowed to be a minor of $G$.
There are many results asserting the existence of tree-decompositions of minimal width which still represent local connectivity properties of the underlying graph, perhaps the best-known being Thomas' theorem that proves for every graph $G$ the existence of a linked tree-decompositon of width tw$(G)$. We prove a general theorem on the existence of linked and lean tree-decompositions, providing a unifying proof of many known results in the field, as well as implying some new results. In particular we prove that every matroid $M$ admits a lean tree-decomposition of width tw$(M)$, generalizing th…
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 different part. Recently, Carmesin, Diestel, Hamann and Hundertmark showed that such a tree-decomposition can be constructed in a canonical way, which makes it invariant under automorphisms of the graph. These canonical tree-decompositions necessarily have parts which contain no tangle of order $k$. We call these parts inessential. Diestel asked what could be s…
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 this $k$-block appears as a part. Carmesin, Diestel, Hamann, Hundertmark and Stein proved that every finite graph has a canonical tree-decomposition of adhesion less than $k$ that distinguishes all its $k$-blocks and tangles of order $k$. We construct such tree-decompositions with the additional property that every separable $k$-block is equal to the unique …
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 following classes has a tree decomposition and a linear-sized path decomposition with bounded intersections: (1) every proper minor-closed class, (2) string graphs with a linear number of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface. Here `linear size' means that the total size of the bags in the path decomposition …
We show that every connected graph has a spanning tree that displays all its topological ends. This proves a 1964 conjecture of Halin in corrected form, and settles a problem of Diestel from 1992.
The tree of decomposition of a $k$-connected graph by a set $\mathfrak S$ of pairwise independent $k$-vertex cutsets is defined as follows. The vertices of this tree are cutsets of $\mathfrak S$ and parts of decomposition of the graph by the set $\mathfrak S$, each cutset is adjacent to all parts that contain it. We prove, that the graph described above is a tree. The tree of decomposition of a biconnected graph is a particular case of this construction: it is the tree of decomposition of a biconnected graph by the set of all its single cutsets (i.e. 2-vertex cutsets, that are independent with…
In answer to a question of Eggleton, we prove that the complete multigraph on 5 vertices with edge multiplicity 6, namely $K_{5}^{(6)}$, has a decomposition into 5 copies of the family of trees of order 5 and that $K_{7}^{(22)}$ has a decomposition into 7 copies of the family of trees of order 7. We prove something similar for $K_{2n+1}$ for $n \le 13$.
We prove that deciding whether the edge set of a graph can be partitionned into two spanning trees with orientation constraints is NP-complete. If P $\neq$ NP then this disproves a conjecture of Recski.
We give a short proof that every finite graph (or matroid) has a tree-decomposition that displays all maximal tangles. This theorem for graphs is a central result of the graph minors project of Robertson and Seymour and the extension to matroids is due to Geelen, Gerards and Whittle.
In this work a composition-decomposition technique is presented that correlates tree eigenvectors with certain eigenvectors of an associated so-called skeleton forest. In particular, the matching properties of a skeleton determine the multiplicity of the corresponding tree eigenvalue. As an application a characterization of trees that admit eigenspace bases with entries only from the set {0, 1,-1} is presented. Moreover, a result due to Nylen concerned with partitioning eigenvectors of tree pattern matrices is generalized.
A separation of a graph $G$ is a partition $(A_1, A_2, C)$ of $V(G)$ such that $A_1$ is anticomplete to $A_2$. A classic result from Robertson and Seymour's Graph Minors Project states that there is a correspondence between tree decompositions and laminar collections of separations. A many-sided separation of a graph $G$ is a partition $(A_1, \ldots, A_k, C)$ of $V(G)$ such that $A_i$ is anticomplete to $A_j$ for all $1 \leq i < j \leq k$. In this note, we show a correspondence between tree decompositions with a certain parity property, called deciduous tree decompositions, and laminar collect…
A graph $G$ is said to be ubiquitous, if every graph $Γ$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every locally finite graph is ubiquitous. In this paper we show that locally finite graphs admitting a certain type of tree-decomposition, which we call an extensive tree-decomposition, are ubiquitous. In particular this includes all locally finite graphs of finite tree-width, and also all locally finite graphs with finitely many ends, all of which have finite degree. It re…
We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem on free Dyck paths and some genera…
The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree depth. In this paper we survey results related to this concept and particularly explain how these graphs are used to decompose and construct more complex graphs and structures. In more technical terms we survey some of the properties and applications of low tree depth decomposition of graphs.
There are different concepts regarding to tree decomposition of a graph $G$. For the Hypercube $Q_n$, these concepts have been shown to have many applications. But some diverse papers on this subject make it difficult to follow what is precisely known. In this note first we will mention some known results on the tree decomposition of hypercubes and then introduce new explicit constructions for the previously known and unknown cases.
We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a 191-edge-connected graph of size divisible by 4, then $G$ has a $Y$-decompo…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph $K_{2n}$, there exists a decomposition of $K_{2n}$ into isomorphic rainbow spanning trees. This settles conjectures of Brualdi--Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.
This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader class, comparable to that given in work of Szegedy, through further iteration.
We present a new method to count unrooted maps on the sphere up to orientation-preserving homeomorphisms. The principle, called tree-decomposition, is to deform a map into an arborescent structure whose nodes are occupied by constrained maps. Tree-decomposition turns out to be very efficient and flexible for the enumeration of constrained families of maps. In this article, the method is applied to count unrooted 2-connected maps and, more importantly, to count unrooted 3-connected maps, which correspond to the combinatorial types of oriented convex polyhedra. Our method improves significantly …
The Dulmage-Mendelsohn decomposition is a classical canonical decomposition in matching theory applicable for bipartite graphs, and is famous not only for its application in the field of matrix computation, but also for providing a prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn decomposition is stated and proved using the two color classes, and therefore generalizing this decomposition for nonbipartite graphs has been a difficult task. In this paper, we obtain a new canonical decomposition that is a generalization of the Dulmage-Mendelsohn decomposition for arbit…
In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. We also give a bijection between the set of functions from $[n+1]$ to $[n]$ and the set of triply rooted trees on $[n]$, which leads to the refined enumeration of functions from $[n+1]$ to $[n]$ with respect to the number of elements in the orbit of $n+1$ and the number of period…
A recent result of Condon, Kim, Kühn and Osthus implies that for any $r\geq (\frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an approximate decomposition into any collections of $n$-vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost $αn$-regular graph $G$ with any $α>0$ and a collection of bounded degree trees on at most $(1-o(1))n$ vertices if $G$ does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into …
Substantial efforts have been made to compute or estimate the minimum number $c(G)$ of cycles needed to partition the edges of an Eulerian graph. We give an equivalent characterization of Eulerian graphs of treewidth $2$ and with maximum degree $4$. This characterization enables us to present a linear time algorithm for the computation of $c(G)$ for all $G$ in this class.
The Tree Decomposition Conjecture by Barát and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of $T$, then $G$ can be edge-decomposed into subgraphs isomorphic to $T$. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from $T$ by vertex-identifications. We…
In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth.
We prove that if a graph has a tree-decomposition of width at most w, then it has a tree-decomposition of width at most w with certain desirable properties. We will use this result in a subsequent paper to show that every 2-connected graph of large path-width has a minor isomorphic to either a large tree with a vertex attached to every vertex of the tree or a large outerplanar graph.
We study the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, providing a new perspective on work of Thomas Lam. This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements. We also show that the affine code readily encodes a number of basic combinatorial properties of an affine permutation. As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which per…
Tree decompositions whose trees are subgraphs: An application of Simon's factorization
LaBRI, Université de Bordeaux（波尔多大学）；Département d’Informatique, Université libre de Bruxelles（布鲁塞尔自由大学）；Department of Theoretical Computer Science, Jagiellonian University（雅盖隆大学）；Faculty of Mathematics, Natural Sciences and Information Technologies and Andrej Marušič Institute, University of Primorska（普里莫斯卡大学）；Institute of Informatics, Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw（华沙大学）
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 divisor of degree $k$ that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality.
For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).
In this paper, we build on recent results by Chauve et al. (2014) and Bahrani and Lumbroso (2017), which combined the split-decomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphs---in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic). Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs, which we describe in more detail further down in this paper…
The modular decomposition of a symmetric map $δ\colon X\times X \to Υ$ (or, equivalently, a set of symmetric binary relations, a 2-structure, or an edge-colored undirected graph) is a natural construction to capture key features of $δ$ in labeled trees. A map $δ$ is explained by a vertex-labeled rooted tree $(T,t)$ if the label $δ(x,y)$ coincides with the label of the last common ancestor of $x$ and $y$ in $T$, i.e., if $δ(x,y)=t(\mathrm{lca}(x,y))$. Only maps whose modular decomposition does not contain prime nodes, i.e., the symbolic ultrametrics, can be exaplained in this manner. Here we co…
The scramble number of a graph is an invariant recently developed to study chip-firing games and divisorial gonality. In this paper we introduce the screewidth of a graph, based on a variation of the existing literature on tree-cut decompositions. We prove that this invariant serves as an upper bound on scramble number, though they are not always equal. We study properties of screewidth, and present results and conjectures on its connection to divisorial gonality.
Tree-cut width is a graph parameter introduced by Wollan that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but for which we can state and prove a tight duality theorem relating it to naturally defined dual objects: appropriately defined brambles and tangles. Using this result we also propose a game characte…
