  The Firefighting problem is defined as follows. At time $t=0$, a fire breaks
out at a vertex of a graph. At each time step $t \geq 0$, a firefighter
permanently defends (protects) an unburned vertex, and the fire then spread to
all undefended neighbors from the vertices on fire. This process stops when the
fire cannot spread anymore. The goal is to find a sequence of vertices for the
firefighter that maximizes the number of saved (non burned) vertices.
  The Firefighting problem turns out to be NP-hard even when restricted to
bipartite graphs or trees of maximum degree three. We study the parameterized
complexity of the Firefighting problem for various structural
parameterizations. All our parameters measure the distance to a graph class (in
terms of vertex deletion) on which the firefighting problem admits a polynomial
time algorithm. Specifically, for a graph class $\mathcal{F}$ and a graph $G$,
a vertex subset $S$ is called a modulator to $\mathcal{F}$ if $G \setminus S$
belongs to $\mathcal{F}$. The parameters we consider are the sizes of
modulators to graph classes such as threshold graphs, bounded diameter graphs,
disjoint unions of stars, and split graphs.
  To begin with, we show that the problem is W[1]-hard when parameterized by
the size of a modulator to diameter at most two graphs and split graphs. In
contrast to the above intractability results, we show that Firefighting is
fixed parameter tractable (FPT) when parameterized by the size of a modulator
to threshold graphs and disjoint unions of stars, which are subclasses of
diameter at most two graphs. We further investigate the kernelization
complexity of these problems to find that firefighting admits a polynomial
kernel when parameterized by the size of a modulator to a clique, while it is
unlikely to admit a polynomial kernel when parameterized by the size of a
modulator to a disjoint union of stars.
