Résumé en anglais

In this paper we study the parameterized complexity of the firefighter problem. More precisely, we show that Saving k -Vertices and its dual Saving All But k -Vertices are both W[1]-hard for parameter k even for bipartite graphs. We also investigate several cases for which the firefighter problem is tractable. For instance, Saving k -Vertices is fixed-parameter tractable on planar graphs for parameter k. Moreover, we prove a lower bound to polynomial kernelization for Saving All But k -Vertices.