Burn and Win.

Given a graph G and an integer k , the Graph Burning problem asks whether the graph G can be burned in at most k rounds. Graph burning is a model for information spreading in a network, where we study how fast the information spreads in the network through its vertices. In each round, the fire is started at an unburned vertex, and fire spreads from every burned vertex to all its neighbors in the subsequent round burning all of them and so on. The minimum number of rounds required to burn the whole graph G is called the burning number of G . Graph Burning is NP-hard even for the union of disjoint paths. Moreover, Graph Burning is known to be W[1]-hard when parameterized by the burning number and para-NP-hard when parameterized by treewidth. In this paper, we prove the following results: In this paper, we give an explicit algorithm for the problem parameterized by treewidth, τ and k , that runs in time k 2 τ 4 k 5 τ n O ( 1 ) . This also gives an FPT algorithm for Graph Burning parameterized by burning number for apex-minor-free graphs. Y. Kobayashi and Y. Otachi [Algorithmica 2022] proved that the problem is FPT parameterized by distance to cographs and gave a double exponential time FPT algorithm parameterized by distance to split graphs. We improve these results partially and give an FPT algorithm for the problem parameterized by distance to cographs ∩ split graphs (threshold graphs) that runs in 2 O ( t ln t ) time. We design a kernel of exponential size for Graph Burning in trees. Furthermore, we give an exact algorithm to find the burning number of a graph that runs in time 2 n n O ( 1 ) , where n is the number of vertices in the input graph.