Homology of graph burnings

TL;DR

This paper introduces a novel algebraic topology framework for graph burnings, defining a configuration space and homology theory that reveal structural properties and relationships between burnings of graphs and their subgraphs.

Contribution

It develops a new categorical and homological approach to analyze graph burnings, including the definition of a configuration space and burning homology.

Findings

The one-dimensional skeleton of the configuration space matches the complement graph of G.

Burning homology captures structural properties of graph burnings.

The framework applies to various examples illustrating its versatility.

Abstract

In this paper we study graph burnings using methods of algebraic topology. We prove that the time function of a burning is a graph map to a path graph. Afterwards, we define a category whose objects are graph burnings and morphisms are graph maps which commute with the time functions of the burnings. In this category we study relations between burnings of different graphs and, in particular, between burnings of a graph and its subgraphs. For every graph, we define a simplicial complex, arising from the set of all the burnings, which we call a configuration space of the burnings. Further, simplicial structure of the configuration space gives burning homology of the graph. We describe properties of the configuration space and the burning homology theory. In particular, we prove that the one-dimensional skeleton of the configuration space of a graph $G$ coincides with the complement graph of $G$. The results are illustrated with numerous examples.