  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.
