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.

中文翻译：

---

防火门问题

假设我们有一个由图 $G$ 表示的网络。潜在的火灾（或其他类型的传染病）可能会在 $G$ 的某个顶点爆发。我们能够通过在 $k$ 的其他 $G$ 顶点处建立防火带来应对这次爆发，以便火不能通过这些强化顶点。现在出现的问题是，假设火灾将蔓延到并非完全位于防火带的 $k$ 个顶点之后的每个顶点，哪个 $k$ 顶点将导致最多数量的顶点从火灾中被救出。这就是{\sc Firebreak}决策问题的本质，也是本文的重点。我们确定该问题在分裂图类和二部图类上都是难以解决的，但是当限于具有恒定有界树宽的图时，可以在线性时间内解决，或在多项式时间内受限于交集图。我们还考虑了一些密切相关的问题。