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Burnings of trees and their homologies

Yuri Muranov, Anna Muranova

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.

Burnings of trees and their homologies

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.
Paper Structure (9 sections, 20 theorems, 50 equations, 6 figures, 1 table)

This paper contains 9 sections, 20 theorems, 50 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notions of the graph theory
  4. Graph burning and corresponding graph filtrations
  5. Burning of a spanning tree
  6. Burning homomorphisms and trees
  7. End time of burnings
  8. Digraphs structure on burned trees
  9. Strong configuration burning space

Key Result

Lemma 2.7

A graph $T=(V_T, E_T)$ is a tree if and only if for any two distinct vertices $v,w\in V_T$ there exists exactly one path $v = v_0, a_1, v_1, a_2, \dots , v_n =~w$.

Figures (6)

  • Figure 1: The tree $T$ for which burning homomorphism does not exist.
  • Figure 2: The ambient tree $Q$ for which burning homomorphism exists.
  • Figure 3: Graphs $Q_1, Q_2,Q_3,Q_4$ for Example \ref{['ex::T15']}
  • Figure 4: Graph $Y_{15}$ for Example \ref{['ex::T15']}
  • Figure 5: Strong configuration space of the graph $Y_8$
  • ...and 1 more figures

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 38 more