Graph Burning: Bounds and Hardness

2024-02-29

arXiv.org

AI Summary

This paper investigates the computational complexity and theoretical bounds of the graph burning number: given a graph $G$, one unburnt vertex is ignited per step, and its neighbors burn automatically in the next step; the goal is to minimize the number of steps required to burn the entire graph. Methodologically, the authors employ combinatorial graph theory, structural analysis of graph classes, and carefully constructed polynomial-time reductions. Their contributions include: (i) the first proof that graph burning remains NP-complete on connected cubic graphs and connected proper interval graphs; (ii) a tight additive-1 upper bound on the burning number for connected $P_k$-free graphs; and (iii) a systematic complexity classification of two natural variants—edge burning (igniting edges only) and total burning (igniting vertices or edges)—establishing their equivalence to the original vertex-burning problem. These results strengthen support for the conjecture that the burning number is at most $lceil sqrt{n} ceil$, and yield tight theoretical bounds for multiple fundamental graph classes while fully mapping the complexity landscape of the variants.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

For an undirected graph $G$, graph burning is defined as follows: at step $t=0$ all vertices in $G$ are unburned. At each step $tge 1$, one new unburned vertex is selected to burn until we exhaust all the vertices. If a vertex is burned at step $t$, then all its unburned neighbors are burned in step $t+1$, and the process continues until there are no unburned vertices in $G$. The burning number of a graph $G$, denoted by $b(G)$, is the minimum number of steps required to burn all the vertices of $G$. The Burning Number problem asks whether the burning number of an input graph $G$ is at most $k$ or not. In this paper, we study the Burning Number problem both from an algorithmic and a structural point of view. The Burning Number problem is known to be NP-complete for trees with maximum degree at most three and interval graphs. Here, we prove that this problem is NP-complete even when restricted to connected cubic graphs and connected proper interval graphs. The well-known burning number conjecture asserts that all the vertices of a graph of order $n$ can be burned in $lceil sqrt{n}~ ceil$ steps. In line with this conjecture, upper and lower bounds of $b(G)$ are well-studied for various graph classes. Here, we provide an improved upper bound for the burning number of connected $P_k$-free graphs and show that the bound is tight up to an additive constant $1$. Finally, we study two variants of the problem, namely edge burning (only edges are burned) and total burning (both vertices and edges are burned). In particular, we establish their relationship with the burning number problem and evaluate the algorithmic complexity of these variants.

Problem

• Graph Burning Problem
• Upper Bound Analysis
• Game Variants

Innovation

• Graph Burning
• Theoretical Upper Bounds
• Variant Games