is allowed to place one firefighter on a vertex of the graph in order to hose that vertex with water, thus preventing it from ever catching fire. The goal of this 'game' is to contain the fire while preventing a maximum number of vertices of the graph from catching fire.
We investigate this problem from a complexity point of view. The starting point of this investigation is the fact that the problem is NP-complete on trees. We then answer two of your most burning questions:
1) Is the problem fixed-parameter tractable and does it have a polynomial kernel?
2) Does the NP-completeness on trees mean that the problem is NP-complete on all sensible graph classes?