DeepAI

# Unfoldings and coverings of weighted graphs

Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings. We prove similar statements for unfoldings. Our study of the difficult proof of Leighton's Theorem lead us to generalize coverings and similarly, unfoldings, by attaching finite or infinite weights to edges of the covered or unfolded graphs. This generalization yields a canonical factorization of the universal covering of any finite graph, that (provably) does not exist without using weights. Introducing infinite weights provides us with finite descriptions of regular trees having nodes of countably infinite degree. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.

05/16/2023

### Decomposition of (infinite) digraphs along directed 1-separations

We introduce torsoids, a canonical structure in matching covered graphs,...
04/06/2019

### X-Ramanujan Graphs

Let X be an infinite graph of bounded degree; e.g., the Cayley graph of ...
10/16/2022

### A Note On ℓ-Rauzy Graphs for the Infinite Fibonacci Word

The ℓ-Rauzy graph of order k for any infinite word is a directed graph i...
08/22/2022

### Digraph Networks and Groupoids

We answer a question posed by Dougherty and Faber in , "Network routi...
01/27/2022

### Positive First-order Logic on Words and Graphs

We study FO+, a fragment of first-order logic on finite words, where mon...
12/04/2019

### A Constructive Formalization of the Weak Perfect Graph Theorem

The Perfect Graph Theorems are important results in graph theory describ...
02/14/2023

### Graph subshifts

We propose a definition of graph subshifts of finite type that can be se...