We consider the complexity of the firefighter problem where a budget of b ≥ 1 fire-fighters are available at each time step. This problem is known to be NP-complete even on trees of degree at most three and b = 1 [14] and on trees of bounded degree (b + 3) for any fixed b ≥ 2 [4]. 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, first, we prove that the problem is NP-complete even on trees of pathwidth at most three for any b ≥ 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 ".