A procedure called \textit{graph burning} was introduced to facilitate the modelling of spread of an alarm, a social contagion, or a social influence or emotion on graphs and networks. Graph burning runs on discrete time-steps (or rounds). At each step $t$, first (a) an unburned vertex is burned (as a \textit{fire source}) from "outside", and then (b) the fire spreads to vertices adjacent to the vertices which are burned till step $t-1$. This process stops after all the vertices of $G$ have been burned. The aim is to burn all the vertices in a given graph in minimum time-steps. The least number of time-steps required to burn a graph is called its \textit{burning number}. The less the burning number is, the faster a graph can be burned. Burning a general graph optimally is an NP-Complete problem. It has been proved that optimal burning of path forests, spider graphs, and trees with maximum degree three is NP-Complete. We study the \textit{graph burning problem} on several sub-classes of \textit{geometric graphs}. We show that burning interval graphs (Section 7.1, Theorem 7.1), permutation graphs (Section 7.2, Theorem 7.2) and disk graphs (Section 7.3, Theorem 7.3) optimally is NP-Complete. In addition, we opine that optimal burning of general graphs (Section 9.2, Conjecture 9.1) cannot be approximated better than 3-approximation factor.

中文翻译：

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燃烧几何图形

引入了一个名为 \textit{graph burn} 的程序，以促进对警报传播、社会传染或社会影响或情感在图和网络上的建模。图刻录在离散的时间步长（或轮次）上运行。在每一步 $t$，首先 (a) 一个未燃烧的顶点从“外部”燃烧（作为 \textit{fire source}），然后（b）火蔓延到与燃烧的顶点相邻的顶点，直到步骤$t-1$。这个过程在 $G$ 的所有顶点都被烧毁后停止。目的是以最小的时间步长烧毁给定图中的所有顶点。刻录图形所需的最少时间步数称为它的 \textit{刻录数}。烧录次数越少，图烧录的速度就越快。最佳地刻录通用图是一个 NP-Complete 问题。已经证明，路径森林、蜘蛛图和最大度数为 3 的树木的最佳燃烧是 NP-Complete。我们在 \textit{geometric graphs} 的几个子类上研究 \textit{graph burn problem}。我们表明燃烧间隔图（第 7.1 节，定理 7.1）、置换图（第 7.2 节，定理 7.2）和磁盘图（第 7.3 节，定理 7.3）是 NP-Complete 最佳的。此外，我们认为一般图的最佳燃烧（第 9.2 节，猜想 9.1）不能比 3-近似因子更好地近似。2) 和磁盘图（第 7.3 节，定理 7.3）最佳地是 NP-Complete。此外，我们认为一般图的最佳燃烧（第 9.2 节，猜想 9.1）不能比 3-近似因子更好地近似。2) 和磁盘图（第 7.3 节，定理 7.3）最佳地是 NP-Complete。此外，我们认为一般图的最佳燃烧（第 9.2 节，猜想 9.1）不能比 3-近似因子更好地近似。