The firefighter problem for cubic graphs

We show that the firefighter problem is NP-complete for cubic graphs. We also show that given a rooted tree of maximum degree three in which every leaf is at the same distance from the root, it is NP-complete to decide whether or not there is a strategy that protects every leaf from the fire, which starts at the root. By contrast, we describe a polynomial time algorithm to decide if it is possible to save a given subset of the vertices of a graph with maximum degree three, provided the fire breaks out at a vertex of degree at most two.