Graph Burning: Bounds and Hardness

Abstract

Graph burning is a discrete-time process that models the propagation of information in a network. Given an undirected graph whose vertices are initially unburned, the process evolves in discrete rounds. At each round, an unburned vertex is selected and burned, while any unburned vertex adjacent to a vertex burned in the previous round also becomes burned. The burning number of a graph is the minimum number of steps to burn all its vertices. The BURNING NUMBER PROBLEM asks whether the burning number of an input graph $G$ is at most $k$. In this paper, we investigate the graph burning problem from both algorithmic and structural viewpoints. Although the problem is known to be NP-complete on interval graphs, we strengthen this result by proving that it remains NP-complete even when restricted to connected proper interval graphs. We also study the burning number of $P_k$-free graphs. Motivated by the well-known burning number conjecture, which states that every connected graph of order $n$ has burning number at most $\lceil \sqrt{n}~\rceil$, we establish an improved upper bound for connected $P_k$-free graphs and show that this bound is tight up to an additive constant of $1$. Finally, we study two variants of the problem: edge burning and total burning. We establish fundamental relationships between these variants and the classical burning, and we determine the computational complexity of the corresponding decision problems.