Ann-Kathrin Elm, Jan Kurkofka
Carmesin has extended Robertson and Seymour's tree-of-tangles theorem to the infinite tangles of locally finite infinite graphs. We extend it further to the infinite tangles of all infinite graphs. Our result has a number of applications for the topology of infinite graphs, such as their end spaces and their compactifications.
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In finite graphs, finite-order tangles offer an abstract description of highly connected substructures. In infinite graphs, infinite-order tangles compactify the graphs in the same way the ends compactify connected locally finite graphs. Thus, the arising tangle compactification extends the well-known Freudenthal compactification from connected locally finite graphs to arbitrary infinite graphs. This Master's thesis investigates the tangle compactifcation.
Tangle structure trees II: trees of tangles and tangle-tree duality
本文通过纠缠结构树推导了两种基本纠缠定理的新版本，扩展了纠缠树定理并展示了如何将非存在性证明转换为树分解。
We show that the tangle space of a graph, which compactifies it, is a quotient of its Stone-Čech remainder obtained by contracting the connected components.
Obtaining trees of tangles from tangle-tree duality
本研究利用抽象分离系统的缠结树对偶定理证明缠结树定理，通过限制节点度强化已有定理，还小幅强化并简化了对偶定理的证明，使其可推导不同阶缠结的缠结树定理。
Tangle-tree duality in infinite graphs
本文将罗伯逊和塞伊姆的网树对偶性定理推广到无限图中。
Tangle structure trees
本文提出纠缠结构树数据结构，能同时展示抽象分离系统中所有F-纠缠，以及证明其不存在或无法扩展的证书。该结构扩展了经典树-纠缠对偶定理至更一般的分离系统。
Refining trees of tangles in abstract separation systems: inessential parts
本文将 Erde 统一图中缠结树定理与缠结-树对偶定理的结果推广到抽象分离系统，以刻画一般离散结构中 $k$-缠结的数量与位置。
Trees of tangles in abstract separation systems
本文针对仅结构子模的抽象分隔系统建立规范与非规范缠结树定理，统一推出图、拟阵等既有结果并显著简化证明。
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 graph that displays, up to a certain natural equivalence, all of the $k$-separations of the matroid or graph that are non-trivial with respect to the tangle.
A canonical tree-of-tangles theorem for structurally submodular separation systems
本文证明结构次模分离系统必存在可典范区分其全部缠结的树集，推广了缠结树定理。
Optimal trees of tangles: refining the essential parts
将Robertson和Seymour的两个固定阶tangle定理统一为一个最优定理，证明每个图G的树分解可精炼为部分极小或足够小的树分解。
We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets. Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width. Our results can be used to derive short proofs of all the classical duality theorems for wi…
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.
Trees of tangles in infinite separation systems
本文将裂割引理推广至无限情形，并据此建立无限图与无限抽象分离系统中的缠结树型定理。
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…
Taffy, Trees, and Tangles
本文探讨了扭绳机、分数树和有理结之间的关系，提出了扭绳类比并建立了有理结与扭绳操作的几何联系。
We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite different. The theorem asserts a duality between the existence of high cohesiveness somewhere local and a global overall tree structure. We describe cohesive substructures in a unified way in the format of tangles: as orientations of low-order separations satisfying certain consistency axioms. These axioms can be expressed without reference to the underlying structure, such as a graph or matroid, bu…
Tanglegrams with a Unique 1-Crossing-Critical Subtanglegram have Tangle Crossing Number 1
本文证明恰好包含一个交叉诱导子纠缠图的纠缠图其纠缠交叉数必为1，并探讨包含k个此类子纠缠图时交叉数是否有界。
Limit-closed Profiles
本文针对无限分离系统提出了一个缠结树定理，其特点是不对分离系统本身施加限制，而是对要在树中排列的缠结施加限制。
A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing, a tanglegram is called planar. We show that every non-planar tanglegram contains one of two non-planar 4-leaf tanglegrams as induced subtanglegram, which parallels Kuratowski's Theorem.
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 neighbourhood precisely equal to $X$. We further provide a concrete separation system whose $\aleph_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\vert G\vert_Γ$ is a quotient of Diestel's (denoted by $\vert G\vert_Θ$), and both use tangles to compactify a graph in much the same way as the ends of a locally finite…
A Note on Generic Tangle Algorithms
本文整理了抽象分离系统中处理 tangle 的三种基础算法，涵盖寻找 tangle、证明其不存在或加速搜索，以及构建 tangle 树。
Efficiently distinguishing all tangles in locally finite graphs
本文证明了没有粗端的局部有限图存在能有效区分所有缠结的规范树分解，并通过分析分离序列极限行为的新方法，在阻碍处展示出粗端。
We show how an image can, in principle, be described by the tangles of the graph of its pixels. The tangle-tree theorem provides a nested set of separations that efficiently distinguish all the distinguishable tangles in a graph. This translates to a small data set from which the image can be reconstructed. The tangle duality theorem says that a graph either has a certain-order tangle or a tree-structure witnessing that this cannot exist. This tells us the maximum resolution at which the image contains meaningful information.
We show that an arbitrary infinite graph can be compactified by its ${\aleph_0}$-tangles in much the same way as the ends of a locally finite graph compactify it in its Freudenthal compactification. In general, the ends then appear as a subset of its ${\aleph_0}$-tangles. The ${\aleph_0}$-tangles of a graph are shown to form an inverse limit of the ultrafilters on the sets of components obtained by deleting a finite set of vertices. The ${\aleph_0}$-tangles that are ends are precisely the limits of principal ultrafilters. The ${\aleph_0}$-tangles that correspond to a highly connected part, or …
We show that, for any graph or matroid, there is a tree that simultaneously distinguishes its maximal tangles, and, for each maximal tangle $\mathcal{T}$ that satisfies an additional robustness condition, displays all of the non-trivial separations relative to $\mathcal{T}$, up to a natural equivalence.
Ends as tangles
本文研究无限图端点所诱导的缠结，刻画了各有限阶闭缠结的特征，并给出其与多数投票顶点集及顶点度数、支配顶点数之间的等价条件。
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 prove a duality theorem applicable to a a wide range of specialisations, as well as to some generalisations, of tangles in graphs. It generalises the classical tangle duality theorem of Robertson and Seymour, which says that every graph either has a large-order tangle or a certain low-width tree-decomposition witnessing that it cannot have such a tangle. Our result also yields duality theorems for profiles and for $k$-blocks. This solves a problem studied, but not solved, by Diestel and Oum and answers an earlier question of Carmesin, Diestel, Hamann and Hundertmark.
Given a graph or a matroid, a tree of tangles is a tree decomposition that displays the structure of the connectivity: every edge of the decomposition tree induces a separation, that is, a way to divide the graph or matroid into two parts; and for every two highly connected areas (encoded as tangles) that live on different sides of some separation, some separation induced by an edge distinguishes them. Separations induced by a tree of tangles cannot cross. One approach to display even more connectivity structure is to insert even more structure into a tree of tangles, for example the flowers t…
We prove a tangle-tree theorem and a tangle duality theorem for abstract separation systems $\vec S$ that are submodular in the structural sense that, for every pair of oriented separations, $\vec S$ contains either their meet or their join defined in some universe $\vec U$ of separations containing $\vec S$. This holds, and is widely used, if $\vec U$ comes with a submodular order function and $\vec S$ consists of all its separations up to some fixed order. Our result is that for the proofs of these two theorems, which are central to abstract tangle theory, it suffices to assume the above str…
Describing ends and tangles (and their edge variants) through Boolean algebras and functors
本文通过布尔代数和Stone对偶，统一了无限图的端空间、缠结空间和边方向空间，并给出了缠结的边类比定义，证明了它们等价于边方向。
Canonical graph decompositions via local separations
本文提出通过局部分离理论实现图的规范分解，解决了传统方法中无限覆盖图不可计算的问题，并证明了DJKK26中的分解可计算性。
Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing "k-connected components" of a graph (though in a twisted way). They play an important role in graph minor theory. An interesting aspect of tangles is that they cannot only be defined for graphs, but more generally for arbitrary connectivity functions (that is, integer-valued submodular and symmetric set functions). However, tangles are difficult to deal with algorithmically. To start with, it is unclear how to represent them, because they are families …
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.
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 for more general combinatorial structures, which has further applications. These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem.
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, H…
Extending the well-known star-comb lemma for infinite graphs, we characterise the graphs that do not contain an infinite comb or an infinite star, respectively, attached to a given set of vertices. We offer several characterisations: in terms of normal trees, tree-decompositions, ranks of rayless graphs and tangle-distinguishing separators.
Entanglements
本文提出一种显式且规范的树分解一步构造法，用于区分图中的所有纠缠，基于局部性质公理化，并推广到局部有限图和拟阵。
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 …
A tangle is a connected topological space constructed by gluing several copies of the unit interval $[0, 1]$. We explore which tangles guarantee envy-free allocations of connected shares for n agents, meaning that such allocations exist no matter which monotonic and continuous functions represent agents' valuations. Each single tangle $\mathcal{T}$ corresponds in a natural way to an infinite topological class $\mathcal{G}(\mathcal{T})$ of multigraphs, many of which are graphs. This correspondence links EF fair division of tangles to EFk$_{outer}$ fair division of graphs. We know from Bilò et a…
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…
Profinite separation systems
本文研究无限分离系统与其诱导的有限分离系统之间的关系，建立profinite分离系统的基础理论，为后续证明无限图和拟阵的tangle型对偶定理奠定框架。
We investigate path-, ray- and branch spaces of trees, certain topological spaces naturally associated with order theoretic trees, and provide topological characterisations for these spaces in terms of the existence of certain kinds of (sub-)bases. Our results yield a solution to a problem due to Diestel from 1992, by establishing the following equivalences for any Hausdorff space $X$: (1) $X$ is homeomorphic to the end space of a graph, (2) $X$ is homeomorphic to the ray space of a special order tree, (3) $X$ admits a nested clopen subbase that is noetherian, hereditarily complete and $σ$-dis…
Plabic Tangles and Cluster Promotion Maps
本文提出Plabic Tangles框架，通过m-VRCs定义promotion映射，并证明其为准簇同构，揭示了与幅度多面体几何及散射幅度奇点的关系。
Representations of infinite tree-sets
本文将有限树集的图论树表示理论推广至无限情形，刻画了可由普通无限树表示的树集，并引入带极限边的树状空间，证明所有无限树集均可由此表示。
We construct a tree T of maximal degree 3 with infinitely many leaves such that whenever finitely many of them are removed, the remaining tree is isomorphic to T. In this sense T resembles an infinite star.
We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting.
Counterexamples regarding linked and lean tree-decompositions of infinite graphs
本文通过构造反例，指出Kriz和Thomas关于树分解的结论无法推广至任意无限树宽的情形，并证明存在局部有限、平面且连通的图不存在精简树分解。
On the locating chromatic number of infinite trees
本文研究了无限树的定位色数，给出了其具有有限定位色数的充要条件，并提出了一种适用于有限树和无限树的定位着色计算算法。
We count the number of closed walks on a vertex in a regular tree using the Catalan's triangle and also the Borel's triangle, showing another combinatorial structure counted by these two array of numbers.
We generalise the construction of infinite matroids from trees of matroids to allow the matroids at the nodes, as well as the field over which they are represented, to be infinite.
A graph $G$ is asymmetrizable if it has a set of vertices whose setwise stablizer only consists of the identity automorphism. The motion $m$ of a graph is the minimum number of vertices moved by any non-identity automorphism. It is known that infinite trees $T$ with motion $m=\aleph_0$ are asymmetrizable if the vertex-degrees are bounded by $2^m.$ We show that this also holds for arbitrary, infinite $m$, and that the number of inequivalent asymmetrizing sets is $2^{|T|}$.
A Cantor-Bernstein-type theorem for spanning trees in infinite graphs
本文证明在无限基数情形下，若图同时具有$\lambda$棵生成树的打包和覆盖，则必可分解为$\lambda$棵生成树；对有限情形仅证明较弱结论。
Gomory and Hu proved that if $ G $ is a finite graph with non-negative weights on its edges, then there exists a tree $ T $ (called now Gomory-Hu tree) on $ V(G) $ such that for all $ u\neq v\in V(G) $ there is an $ e\in E(T) $ such that the two components of $ T-e $ determines an optimal (minimal valued) cut between $ u $ an $ v $ in $ G $. In this paper we extend their result to infinite weighted graphs with finite total weight. Furthermore, we show by an example that one can not omit the condition of finiteness of the total weight.
Halin proved in 1978 that there exists a normal spanning tree in every connected graph $G$ that satisfies the following two conditions: (i) $G$ contains no subdivision of a `fat' $K_{\aleph_0}$, one in which every edge has been replaced by uncountably many parallel edges; and (ii) $G$ has no $K_{\aleph_0}$ subgraph. We show that the second condition is unnecessary.
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.
An infinite family of trees with irreducible characteristic polynomials
研究通过在特定路径顶点附加叶子得到的树，利用数论论证及相关结果，证明当n满足条件时其特征多项式不可约，从而得到无限多个具有不可约特征多项式的非同构树，解决了相关猜想。
We prove local convergence results of rerooted conditioned multi-type Galton--Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine.
$p$-Modulus on radially symmetric trees
本文建立无限根树上无限路径族的 $p$-模理论，证明其可由截断树的 $p$-模极限刻画，并揭示其与最小割、有效电阻及最短路径的联系，还给出径向对称无限二叉树的临界 $p$ 值。
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