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.
翻译:暂无翻译