  The Firefighter problem is to place firefighters on the vertices of a graph
to prevent a fire with known starting point from lighting up the entire graph.
In each time step, a firefighter may be permanently placed on an unburned
vertex and the fire spreads to its neighborhood in the graph in so far no
firefighters are protecting those vertices. The goal is to let as few vertices
burn as possible. This problem is known to be NP-complete, even when restricted
to bipartite graphs or to trees of maximum degree three. Initial study showed
the Firefighter problem to be fixed-parameter tractable on trees in various
parameterizations. We complete these results by showing that the problem is in
FPT on general graphs when parameterized by the number of burned vertices, but
has no polynomial kernel on trees, resolving an open problem. Conversely, we
show that the problem is W[1]-hard when parameterized by the number of unburned
vertices, even on bipartite graphs. For both parameterizations, we additionally
give refined algorithms on trees, improving on the running times of the known
algorithms.
