We consider the complexity of the firefighter problem where a budget of b ≥ 1 firefighters are available at each time step. This problem is proved to be NP-complete even on trees of degree at most three and b = 1 [10] and on trees of bounded degree 6+3 for any fixed b ≥ 2 [3]. In this paper, we provide further insight into the complexity landscape of the problem by showing a complexity dichotomy result with respect to the parameters pathwidth and maximum degree of the input graph. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any 6 ≥ 1. Then we show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph. Finally, we show that the problem remains NP-complete on very dense graphs, namely co-bipartite graphs, but is fixed-parameter tractable with respect to the parameter "cluster vertex deletion".
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