Hardness of Burning Number Problem on Regular Graphs

2026-05-14

AI Summary

This study investigates the computational complexity of the graph burning problem on regular graphs, which asks whether a given graph can be completely “burned” within at most $k$ time steps. For the first time, it is shown that this problem is NP-complete and APX-hard even when restricted to connected cubic graphs and, more generally, to connected $d$-regular graphs for any fixed degree $d \geq 4$. Through carefully designed polynomial-time reductions and rigorous approximation hardness analysis, the work establishes that the burning number problem is not only intractable to solve exactly but also resists efficient approximation on these graph classes. These results substantially advance the theoretical understanding of the computational difficulty inherent in network propagation models.

Abstract

The Burning Number Problem (BNP) models the spread of information or contagion in a network through a discrete-time process on a graph. At each step, one new vertex is selected as a burning source, while fire simultaneously spreads from previously burned vertices to their neighbors. The burning number of a graph is the minimum number of steps required to burn all vertices. The decision version asks whether the burning number is at most a given integer $k$. BNP is known to be NP-complete even on restricted graph classes such as path forests. We study BNP on connected regular graphs, a natural and previously unexplored graph class. We prove that BNP is NP-complete on connected cubic graphs, and moreover APX-hard under this restriction. We further show that BNP remains APX-hard on connected $d$-regular graphs for every fixed $d \geq 4$.

Problem

• Burning Number Problem
• regular graphs
• NP-completeness
• APX-hardness
• cubic graphs

Innovation

• Burning Number Problem
• NP-completeness
• APX-hardness
• regular graphs
• cubic graphs