# The Firebreak Problem

## TL;DR

This paper studies the Firebreak problem in graphs, analyzing its computational complexity and providing efficient algorithms for specific graph classes to optimize fire containment strategies.

## Contribution

It proves the problem's intractability on split and bipartite graphs, and offers linear and polynomial-time solutions for graphs with bounded treewidth and intersection graphs.

## Key findings

- NP-hard on split graphs
- NP-hard on bipartite graphs
- Linear time algorithm for bounded treewidth graphs

## Abstract

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.