The burning number of a graph $G$, denoted by $b(G)$, is the minimum number of steps required to burn all the vertices of a graph where in each step the existing fire spreads to all the adjacent vertices and one additional vertex can be burned as a new fire source. In this paper, we study the burning number problem both from an algorithmic and a structural point of view. The decision problem of computing the burning number of an input graph is known to be NP-Complete for trees with maximum degree at most three and interval graphs. Here, we prove that this problem is NP-Complete even when restricted to connected proper interval graphs and connected cubic graphs. The well-known burning number conjecture asserts that all the vertices of any graph of order $n$ can be burned in $\lceil \sqrt{n}~\rceil$ steps. In line with this conjecture, upper and lower bounds of $b(G)$ are well-studied for various special 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 complexity of these variants.

翻译：图$G$的燃烧数$b(G)$定义为燃烧图中所有顶点所需的最少步数，其中每一步中现有的火势会蔓延至所有相邻顶点，且可额外点燃一个顶点作为新的火源。本文从算法与结构两个角度研究燃烧数问题。已知输入图的燃烧数判定问题在最大度不超过3的树和区间图上为NP完全问题。本文证明该问题在连通真区间图和连通三次图上仍为NP完全问题。著名的燃烧数猜想断言，任意$n$阶图的所有顶点可在$\lceil \sqrt{n}~\rceil$步内燃烧。基于该猜想，各类特殊图类的$b(G)$上下界已被广泛研究。本文给出了连通$P_k$-free图燃烧数的改进上界，并证明该上界在附加常数$1$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）和全燃烧（同时燃烧顶点与边），建立了它们与燃烧数问题的关系，并评估了这些变体的复杂度。