Hardness of Burning Number Problem on Regular Graphs

TL;DR

This paper investigates the computational complexity of the Burning Number Problem on regular graphs, proving NP-completeness and APX-hardness for various classes, including cubic and higher-degree regular graphs.

Contribution

It establishes the NP-completeness and APX-hardness of BNP on connected regular graphs, a previously unexplored class, extending known hardness results.

Findings

BNP is NP-complete on connected cubic graphs.

BNP is APX-hard on connected $d$-regular graphs for all fixed $d \\geq 4.

BNP remains hard even on restricted regular graph classes.

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 $d \geq 4$.