We consider the complexity of the firefighter problem where ${b \geq 1}$ firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al. 2007) and on trees of bounded degree $b+3$ for any fixed budget $b \geq 2$ (Bazgan et al. 2012).
In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of the input graph govern its complexity. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget $b \geq 1$. We then 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.

We consider the complexity of the firefighter problem where ${b \geq 1}$ firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al. 2007) and on trees of bounded degree $b+3$ for any fixed budget $b \geq 2$ (Bazgan et al. 2012).
In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of a graph govern the complexity of the problem. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget $b \geq 1$. We then 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.