Graph Burning: Bounds and Hardness

TL;DR

This work advances graph burning by proving that the Burning Number problem remains NP-complete even for connected proper interval graphs, and by providing a near-tight upper bound $b(G) \le \lceil*\rceil{\frac{k+1}{2}}$ for connected $P_k$-free graphs, with algorithmic implications. It also investigates two natural variants, edge burning and total burning, establishing their relationships to the standard burning number via line and total graphs, deriving tight bounds such as $b(G)-1 \le b_L(G) \le b(G)+1$ and $b(G) \le b_T(G) \le b(G)+1$, proving stronger results for trees, and proving NP-completeness for line graphs of caterpillars and total graphs of bounded-degree trees. The results collectively deepen understanding of contagion-like spreading on graph classes, linking structural properties to computational hardness and enabling targeted algorithmic approaches for restricted graph families.

Abstract

Let $G=(V,E)$ be an undirected graph. The graph burning is defined as follows: at time $t=0$, all vertices in $G$ are unburned. For each time $t\geq 1$, an unburned vertex is chosen to burn, and at each subsequent time, the fire spreads from each burned vertex to all its neighbors. Once a vertex is burned, it remains burned for all future steps. The process continues until all vertices in $V$ are burned. The burning number of a graph $G$, denoted $b(G)$, is the smallest integer $k$ such that there exists a sequence of vertices $(v_1,v_2,\ldots, v_k)\subseteq V$, where $v_i$ is burned at time $i$, and all vertices in $V$ are burned within time step $k$. 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 interval graphs. Here, we prove that this problem is NP-complete even when restricted to 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 $\lceil \sqrt{n}~\rceil$ steps. In line with this conjecture, the 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.

Figures (5)

• Figure 1: An illustration of Construction \ref{['cons:proper']} corresponding to the instance $X=~\{10,11,12,14,15,16\}$ with $m=16$ of the Distinct 3-partition problem. The parallel lines represent the induced path $P$ on $G_p$. The hanging vertices adjacent to the vertices on $P$ represent the vertices $q_{ij}$, for $1\leq i\leq m+1$, $1\leq j\leq i'-1$, and $i'=|Q_i|$.
• Figure 2: Structure of a $Q_i^p$ with 15 vertices in $Q_i$. The dashed line represents the subpaths that are connected to $Q_i^p$ on both ends.
• Figure 3: Tight examples for Theorem \ref{['thm:PkFreeBound']}
• Figure 4: An example of a tree with the corresponding graphs $T_1$ and $T_2$.
• Figure 5: An example of a spike graph and its total graph

Theorems & Definitions (37)

• Conjecture 1.1: Bonato2016Burning
• Theorem 1.2: Bonato2016Burning
• Proposition 3.1: Kare2019ParamBurningAlgo
• Lemma 3.3
• proof
• proof
• Lemma 3.6
• proof
• Lemma 3.7
• proof