arXiv subjects
Publications and source records attributed to Maximilian Teegen.
Applications of tangles of connectivity systems suggest a duality between these, in which for two sets $X$ and $Y\!$ the elements $x$ of $X$ map to subsets $Y_x$ of $Y\!$, and the elements $y$ of $Y\!$ map to subsets $X_y$ of $X$, so that $x\in X_y$ if and only if $y\in Y_x$ for all $x\in X$ and $y\in Y\!$. We explore this duality, and relate the tangles arising from the dual systems to each other.
A set of vertices in a graph is agile if, however we partition the set into two parts, we can always find two vertex-disjoint connected subgraphs where one covers the first and the other the second part. We present a characterization for the existence of large agile sets in terms of $K_{2,k}$ and large strip minors.
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 remains an open question whether every locally finite graph admits an extensive tree-decomposition.
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.
We analyse various structural and order-theoretical aspects of abstract separation systems and partial lattices, as well as the relationship between the different submodularity conditions one can impose on them.
We identify and study a simple combinatorial problem that is derived from submodularity issues encountered in the theory of tangles of graphs and abstract separation systems.
A graph $G$ is said to be $\preceq$-ubiquitous, where $\preceq$ is the minor relation between graphs, if whenever $Γ$ is a graph with $nG \preceq Γ$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \preceq Γ$, where $αG$ is the disjoint union of $α$ many copies of $G$. A well-known conjecture of Andreae is that every locally finite connected graph is $\preceq$-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph~$G$ which implies that $G$ is $\preceq$-ubiquitous. In particular this implies that the full grid is $\preceq$-ubiquitous.
We demonstrate the versatility of the tangle-tree duality theorem for abstract separation systems by using it to prove tree-of-tangles theorems. This approach allows us to strengthen some of the existing tree-of-tangles theorems by bounding the node degrees in them. We also present a slight strengthening and simplified proof of the duality theorem, which allows us to derive a tree-of-tangles theorem also for tangles of different orders.
Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection of cuts of any dataset, tangles aggregate these cuts to point in the direction of a dense structure. As a result, a cluster is softly characterized by a set of consistent pointers. This highly flexible approach can solve clustering problems in various setups, ranging from questionnaires over community detection in graphs to clustering points in metric spaces. The output of our proposed framework is hierarchical and induces the notion of a soft dendrogram, which can help explore the cluster structure of a dataset. The computational complexity of aggregating the cuts is linear in the number of data points. Thus the bottleneck of the tangle approach is to generate the cuts, for which simple and fast algorithms form a sufficient basis. In our paper we construct the algorithmic framework for clustering with tangles, prove theoretical guarantees in various settings, and provide extensive simulations and use cases. Python code is available on github.
In this note we gather the theoretical outlines of three basic algorithms for tangles in abstract separation systems: a naive tree search for finding tangles; an algorithm which outputs a certificate for the non-existence of tangles if possible, and otherwise a way to jump-start the naive tree search; and a way to obtain a tree-of-tangles.
We present infinite analogues of our splinter lemma from [Trees of tangles in abstract separation systems, arXiv:1909.09030]. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite abstract separation systems.
We prove canonical and non-canonical tree-of-tangles theorems for abstract separation systems that are merely structurally submodular. Our results imply all known tree-of-tangles theorems for graphs, matroids and abstract separation systems with submodular order functions, with greatly simplified and shortened proofs.
We show that, given a $ k $-tangle $ \tau $ in a graph $ G $, there always exists a weight function $ w\colon V(G)\to\mathbb{N} $ such that a separation $ (A,B) $ of $ G $ of order $ {<}k $ lies in $ \tau $ if and only if $ w(A)<w(B) $, where $ w(U) := \sum_{u\in U}w(u) $ for $ U\subseteq V(G) $. We show that the same result holds also for tangles of hypergraphs as well as for edge-tangles of graphs, but not for edge-tangles of hypergraphs.
Let $\triangleleft$ be a relation between graphs. We say a graph $G$ is \emph{$\triangleleft$-ubiquitous} if whenever $Γ$ is a graph with $nG \triangleleft Γ$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \triangleleft Γ$, where $αG$ is the disjoint union of $α$ many copies of $G$. The \emph{Ubiquity Conjecture} of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper, which is the first of a series of papers making progress towards the Ubiquity Conjecture, we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.