The burning number of a graph was recently introduced by Bonato et al. Although they mention that the burning number generalises naturally to directed graphs, no further research on this has been done. Here, we introduce graph burning for directed graphs, and we study bounds for the corresponding burning number and the hardness of finding this number. We derive sharp bounds from simple algorithms and examples. The hardness question yields more surprising results: finding the burning number of a directed tree is NP-hard, but FPT; however, it is W[2]-complete for DAGs. Finally, we give a fixed-parameter algorithm to find the burning number of a digraph, with a parameter inspired by research in phylogenetic networks.

中文翻译：

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有向图的燃烧数：边界和计算复杂性

Bonato 等人最近引入了图的燃烧数。尽管他们提到燃烧数自然地推广到有向图，但尚未对此进行进一步研究。在这里，我们介绍了有向图的图燃烧，我们研究了相应燃烧数的界限和找到这个数的难度。我们从简单的算法和示例中得出明确的界限。硬度问题产生了更令人惊讶的结果：找到有向树的燃烧数是 NP-hard，但 FPT；但是，对于 DAG，它是 W[2] 完备的。最后，我们给出了一个固定参数算法来找到有向图的燃烧数，其参数受到系统发育网络研究的启发。