74,440 research outputs found
The Burning Number of Directed Graphs: Bounds and Computational Complexity
The burning number of a graph was recently introduced by Bonato et al. Although they mention that the burning number generalizes 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
The Burning Number of Directed Graphs: Bounds and Computational Complexity
The burning number of a graph was recently introduced by Bonato et al. Although they mention that the burning number generalizes 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
A survey of graph burning
Graph burning is a deterministic, discrete-time process that models how
influence or contagion spreads in a graph. Associated to each graph is its
burning number, which is a parameter that quantifies how quickly the influence
spreads. We survey results on graph burning, focusing on bounds, conjectures,
and algorithms related to the burning number. We will discuss state-of-the-art
results on the burning number conjecture, burning numbers of graph classes, and
algorithmic complexity. We include a list of conjectures, variants, and open
problems on graph burning
A survey of graph burning
Graph burning is a deterministic, discrete-time process that models how influence or contagion spreads in a graph. Associated to each graph is its burning number, which is a parameter that quantifies how quickly the influence spreads. We survey results on graph burning, focusing on bounds, conjectures, and algorithms related to the burning number. We will discuss state-of-the-art results on the burning number conjecture, burning numbers of graph classes, and algorithmic complexity. We include a list of conjectures, variants, and open problems on graph burning
Fronts in reactive convection: bounds, stability and instability
We consider front propagation in a reactive Boussinesq system in an infinite
vertical strip. We establish nonlinear stability of planar fronts for narrow
domains when the Rayleigh number is not too large. Planar fronts are shown to
be linearly unstable with respect to long wavelength perturbations if the
Rayleigh number is sufficiently large. We also prove uniform bounds on the bulk
burning rate and the Nusselt number in the KPP reaction case
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