Burnings of trees and their homologies

TL;DR

This paper explores a novel algebraic topology approach to graph burning, focusing on trees and their homologies, and introduces new concepts like strong burning homology and burning configuration spaces.

Contribution

It extends algebraic topology methods to a new burning process model, establishing relations between graph and spanning tree burnings and defining new homological structures.

Findings

Burning of a tree induces a digraph structure.

Relations between graph and spanning tree burnings are established.

Introduction of strong burning homology and configuration spaces.

Abstract

The problem of graph burning was firstly introduced as a model for different processes of social and network interactions. Recently, the authors of the present paper developed methods of algebraic topology for investigation of this problem. This approach is based on the new definition of burning process which excludes the possibility to choose at any moment vertex for burning from the set of vertices which are already burned at this moment. In this paper we continue to study such burning process using algebraic topology methods. We prove the result about relations between burnings of a graph and burnings of its spanning trees that is similar to the classical case. Afterwards, we describe properties of trees burnings. In particular, we prove that a burning of a tree defines a structure of a digraph on the tree and investigate this structure. We introduce and study a strong burning configuration space of a graph and new strong burning homology which are similar to burning homology defined in our previous paper, but arise from burning homomorphism.