The Firebreak Problem

Suppose we have a network that is represented by a graphG. Potentially a fire (or other type of contagion) might erupt at some vertex ofG. We are able to respond to this outbreak by establishing a firebreak atkother vertices ofG, so that the fire cannot pass through these fortified vertices. The question that now arises is whichkvertices 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 thekvertices of the firebreak. This is the essence of theFirebreakdecision 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.