Abstract
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 with one indegree-0 node 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.
Highlights
The burning number of a graph was recently introduced by Bonato et al [BJR14] as a model of social contagion
Social contagion is the spread of rumours, behaviour, emotions, or other social information through social networks (e.g., [BFJ+12, KGH14])
The topic seems relevant, for example in the same setting as undirected graphs, social contagion: some communication occurs in mostly one direction, such as when people follow other people on social media
Summary
The burning number of a graph was recently introduced by Bonato et al [BJR14] as a model of social contagion. In this paper, we introduce the directed version of graph burning and we study some basic problems related to graph burning. All neighbours of all burned nodes are burned, and one extra node is chosen that is burned as well This models social contagion in the sense that neighbours start to burn correspond to information being spread through the network. The basic questions we study relate to bounds on the burning number for different classes of graphs, and to the computational complexity of determining the burning number of a graph. Both of these problems have been studied for undirected graphs [LL16, BBJ+18, BL19, BBJ+17].
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