Graph burning is a discrete-time process that models the propagation of information in a network. Initially, we have an undirected graph of unburned vertices. At each time step, an unburned vertex is chosen to burn; additionally, unburned vertices with at least one burned neighbor from the previous step also become burned. Once a vertex is burned, it remains burned for all future steps. The burning number of a graph is the minimum number of steps to burn all the vertices of the graph. 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 viewpoint. 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 the burning number of a connected graph of order $n$ is at most $\lceil \sqrt{n}~\rceil$. In line with this conjecture, the upper and lower bounds of the burning number 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 of $1$. Finally, we study two variants of the problem: edge burning and total burning. We establish their relationship with the classical burning and evaluate the algorithmic complexity of these variants.

翻译：图燃烧是一种模拟网络中信息传播的离散时间过程。初始时，我们有一个包含未燃烧顶点的无向图。在每个时间步，选择一个未燃烧顶点进行点燃；此外，若未燃烧顶点在前一步中至少有一个已燃烧的邻居，则该顶点也会被点燃。一旦顶点被点燃，它在所有后续步骤中保持燃烧状态。图的燃烧数是指燃烧所有顶点所需的最小步数。燃烧数问题询问输入图$G$的燃烧数是否最多为$k$。本文从算法和结构两个角度研究燃烧数问题。已知燃烧数问题在区间图上为NP完全问题。本文证明，即使限制在连通真区间图上，该问题仍是NP完全问题。著名的燃烧数猜想断言，$n$阶连通图的燃烧数最多为$\lceil \sqrt{n}~\rceil$。基于这一猜想，各类图类的燃烧数上下界已被广泛研究。本文给出了连通$P_k$-无图燃烧数的改进上界，并证明该界在加法常数$1$的范围内是紧的。最后，我们研究了该问题的两个变体：边燃烧和总燃烧。我们建立了它们与经典燃烧的关系，并评估了这些变体的算法复杂度。