Related papers: Modeling rooted in-trees by finite p-groups
Recent examples of periodic bifurcations in descendant trees of finite p-groups with p in {2,3} are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p-class group of type (2,2,2), resp. (3,3),…
Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups…
Theoretical background and an implementation of the (p)-group generation algorithm by Newman and O'Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite…
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…
We prove that every graph which admits a tree-decomposition into finite parts has a rooted tree-decomposition into finite parts that is linked, tight and componental. As an application, we obtain that every graph without half-grid minor has…
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…
We consider the Bernoulli bond percolation process (with parameter $p$) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric…
Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded…
Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…
This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This…
Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…
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…
A graph class is monotone if it is closed under taking subgraphs. It is known that a monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a so-called tripod, that is, a…
We develop a structural approach to simultaneous embeddability in temporal sequences of graphs, inspired by graph minor theory. Our main result is a classification theorem for 2-connected temporal sequences: we identify five obstruction…
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…
This paper investigates the algebraic and graphical structure of the ring $\mathbb{Z}_{sp}$, with a focus on its decomposition into finite fields, kernels, and special subsets. We establish classical isomorphisms between $\mathbb{F}_s$ and…
We prove that the edge-end space of an infinite graph is metrizable if and only if it is first-countable. This strengthens a recent result by Aurichi, Magalhaes Jr.\ and Real (2024). Our central graph-theoretic tool is the use of tree-cut…
In this paper, we establish the theory of $p$-modulus of a family of infinite paths on an infinite-rooted tree and then explore its interpretation and properties. One key result is the formulation of $p$-modulus on the infinite tree as a…
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…
We provide a finite equational presentation of graphs of treewidth at most three, solving an instanceof an open problem by Courcelle and Engelfriet. We use a syntax generalising series-parallel expressions, denoting graphs with a small…