  Suppose we have a network that is represented by a graph $G$. Potentially a
fire (or other type of contagion) might erupt at some vertex of $G$. We are
able to respond to this outbreak by establishing a firebreak at $k$ other
vertices of $G$, so that the fire cannot pass through these fortified vertices.
The question that now arises is which $k$ vertices will result in the greatest
number of vertices being saved from the fire, assuming that the fire will
spread to every vertex that is not fully behind the $k$ vertices of the
firebreak. This is the essence of the {\sc Firebreak} decision problem, which
is the focus of this paper. We establish that the problem is intractable on the
class of split graphs as well as on the class of bipartite graphs, but can be
solved in linear time when restricted to graphs having constant-bounded
treewidth, or in polynomial time when restricted to intersection graphs. We
also consider some closely related problems.
