Related papers: Representations of infinite tree-sets
We give exact formulas for the transmission (i.e. the sum of all distances between vertices) of perfect trees and rooted powers of (connected finite) graphs.
Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…
We present in this paper a first-order axiomatization of an extended theory $T$ of finite or infinite trees, built on a signature containing an infinite set of function symbols and a relation $\fini(t)$ which enables to distinguish between…
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 give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We…
In this series, we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects such as undirected graphs, directed graphs, bidirected graphs, hypergraphs and finitary matroids.…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
We extend to infinite graphs the matroidal characterization of finite graph duality, that two graphs are dual iff they have complementary spanning trees in some common edge set. The naive infinite analogue of this fails. The key in an…
We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum…
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…
We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their…
The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Fr\"uhwirth, we consider an extension of this theory with an additional…
In evolutionary studies it is common to use phylogenetic trees to represent the evolutionary history of a set of species. However, in case the transfer of genes or other genetic information between the species or their ancestors has…
We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is…
This paper, dating from May 1991, contains preliminary (and unpublishable) notes on investigations about iteration trees. They will be of interest only to the specialist. In the first two sections I define notions of support and embeddings…
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…
Tree representations of (sets of) symmetric binary relations, or equivalently edge-colored undirected graphs, are of central interest, e.g.\ in phylogenomics. In this context symbolic ultrametrics play a crucial role. Symbolic ultrametrics…
To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the…
We give a combinatorial description of shape theory using finite topological $T_0$-spaces (finite partially ordered sets). This description may lead to a sort of computational shape theory. Then we introduce the notion of core for inverse…
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…