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Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…

Logic in Computer Science · Computer Science 2023-06-22 Bruno Courcelle

Hierarchical structure and repetition are prevalent in graphs originating from nature or engineering. These patterns can be represented by a class of parametric-structure graphs, which are defined by templates that generate structure by way…

Data Structures and Algorithms · Computer Science 2020-11-16 Tal Ben-Nun , Lukas Gianinazzi , Torsten Hoefler , Yishai Oltchik

We give a new graph-theoretic proof of Cobham's Theorem which says that the support of an automatic sequence is either sparse or grows at least like $N^\alpha$ for some $\alpha > 0$. The proof uses the notions of tied vertices and cycle…

Combinatorics · Mathematics 2024-05-21 Mieke Wessel

Four algorithms giving rise to graceful graphs from a known (non)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and…

Combinatorics · Mathematics 2020-06-09 Suryaprakash Nagoji Rao

Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a…

Combinatorics · Mathematics 2024-10-10 Rajko Nenadov , Huy Tuan Pham

Graph reachability is the task of understanding whether two distinct points in a graph are interconnected by arcs to which in general a semantic is attached. Reachability has plenty of applications, ranging from motion planning to routing.…

Artificial Intelligence · Computer Science 2025-03-26 Davide Di Pierro , Stephan Mennicke , Stefano Ferilli

Growing graphs describe a multitude of developing processes from maturing brains to expanding vocabularies to burgeoning public transit systems. Each of these growing processes likely adheres to proliferation rules that establish an…

Quantitative Methods · Quantitative Biology 2020-05-27 Ann Sizemore Blevins , Danielle S. Bassett

This paper contains a classification of countable lower 1-transitive linear orders. The notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and is essential for the structure theory of 1-transitive trees.…

Combinatorics · Mathematics 2015-10-22 Silvia Barbina , Katie Chicot

A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite…

Combinatorics · Mathematics 2021-02-11 Yibo Gao , Junyao Peng

We give a direct polynomial-time reduction from parity games played over the configuration graphs of collapsible pushdown systems to safety games played over the same class of graphs. That a polynomial-time reduction would exist was known…

Logic in Computer Science · Computer Science 2018-07-06 Matthew Hague , Roland Meyer , Sebastian Muskalla , Martin Zimmermann

The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic…

Logic in Computer Science · Computer Science 2008-10-29 Dietrich Kuske , Markus Lohrey

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of…

Combinatorics · Mathematics 2018-09-10 Russell Lyons

Trees are fundamental data structure for many areas of computer science and system engineering. In this report, we show how to ensure eventual consistency of optimistically replicated trees. In optimistic replication, the different replicas…

Data Structures and Algorithms · Computer Science 2012-01-10 Stéphane Martin , Mehdi Ahmed-Nacer , Pascal Urso

The ability to adequately model risks is crucial for insurance companies. The method of "Copula-based hierarchical risk aggregation" by Arbenz et al. offers a flexible way in doing so and has attracted much attention recently. We briefly…

Risk Management · Quantitative Finance 2015-06-22 Fabio Derendinger

We consider the notion of herdability, a set-based reachability condition, which asks whether the state of a system can be controlled to be element-wise larger than a non-negative threshold. First a number of foundational results on…

Systems and Control · Computer Science 2019-04-19 Sebastian F Ruf , Magnus Egerstedt , Jeff S. Shamma

Controlling real-world networked systems, including ecological, biomedical, and engineered networks that exhibit higher-order interactions, remains challenging due to inherent nonlinearities and large system scales. Despite extensive…

Optimization and Control · Mathematics 2026-03-23 Joshua Pickard , Xin Mao , Can Chen

This paper describes a design that can be used for Explainable AI. The lower level is a nested ensemble of patterns created by self-organisation. The upper level is a hierarchical tree, where nodes are linked through individual concepts, so…

Artificial Intelligence · Computer Science 2020-11-30 Kieran Greer

Phylogenetic trees constitute an interesting class of objects for stochastic processes due to the non-standard nature of the space they inhabit. In particular, many statistical applications require the construction of Markov processes on…

Probability · Mathematics 2024-10-24 Rodrigo B. Alves , Yuri F. Saporito , Luiz M. Carvalho

We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of…

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special…