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：令 $G=(V,E)$ 为一个无向图。图燃烧过程定义如下：在时刻 $t=0$，图 $G$ 中所有顶点均未燃烧。对于每个时刻 $t\geq 1$，选择一个未燃烧顶点进行燃烧；随后在每个后续时刻，火焰从每个已燃烧顶点蔓延至其所有邻居顶点。一旦顶点被燃烧，其将在后续所有时间步中保持燃烧状态。该过程持续进行，直至 $V$ 中所有顶点均被燃烧。图 $G$ 的燃烧数，记为 $b(G)$，是指满足以下条件的最小整数 $k$：存在顶点序列 $(v_1,v_2,\ldots, v_k)\subseteq V$，其中 $v_i$ 在时刻 $i$ 被燃烧，且 $V$ 中所有顶点在时刻 $k$ 之前均被燃烧。燃烧数判定问题要求判断给定输入图 $G$ 的燃烧数是否不超过 $k$。本文从算法与结构两个角度研究燃烧数问题。已知该问题在区间图上具有 NP 完全性。本文进一步证明，即使限制在连通真区间图上，该问题仍然是 NP 完全的。著名的燃烧数猜想断言：任意 $n$ 阶图的所有顶点均可在 $\lceil \sqrt{n}~\rceil$ 步内被燃烧。围绕该猜想，各类图结构的 $b(G)$ 上下界已得到广泛研究。本文针对连通 $P_k$-free 图改进了燃烧数的上界，并证明该界在可加常数 $1$ 范围内是紧的。最后，我们研究了该问题的两个变体：边燃烧（仅燃烧边）与全燃烧（同时燃烧顶点与边）。特别地，我们建立了它们与燃烧数问题的关联，并评估了这些变体的算法复杂性。