Parameterized Complexity of Graph Burning

Abstract

Graph Burning asks, given a graph G = (V,E) and an integer k, whether there exists (b_{0},dots ,b_{k-1}) in V^{k} such that every vertex in G has distance at most i from some b_{i}. This problem is known to be NP-complete even on connected caterpillars of maximum degree 3. We study the parameterized complexity of this problem and answer all questions by Kare and Reddy [IWOCA 2019] about the parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by k and that it does not admit a polynomial kernel parameterized by vertex cover number unless mathrm {NP} subseteq mathrm {coNP/poly}. We also show that the problem is fixed-parameter tractable parameterized by clique-width plus the maximum diameter among all connected components. This implies the fixed-parameter tractability parameterized by modular-width, by treedepth, and by distance to cographs. Using a different technique, we show that parameterization by distance to split graphs is also tractable. We finally show that the problem parameterized by max leaf number is XP.