In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph and then firefighters protect vertices. On each subsequent turn, the fire spreads to the collective unburnt neighbourhood of all the burning vertices and the firefighters again protect vertices. Once a vertex has been burnt or protected it remains that way for the rest of the game. A common objective with respect to some infinite graph is to determine how many firefighters are necessary to stop the fire from spreading after a finite number of turns, commonly referred to as containing the fire. We introduce the concept of distance-restricted firefighting where the firefighters’ movement is restricted so they can only move up to some fixed distance per turn rather than being able to move without restriction. We establish some general properties of this new game in contrast to properties of the original game, and we investigate specific cases of the distance-restricted game on the infinite square, strong, and hexagonal grids. We conjecture that two firefighters are insufficient on the infinite square grid when , and we pose some questions about how many firefighters are required in general when .

In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph G and then b firefighters protect b vertices. On each subsequent turn, the fire spreads to the collective unburned neighbourhood of all the burning vertices and the firefighters again protect b vertices. Once a vertex has been burned or protected it remains that way for the rest of the game. In distance-restricted firefighting the firefighters’ movement is restricted so they can only move up to some fixed distance d and they may or may not be permitted to move through burning vertices. In this paper we establish the NP-completeness of the distance-restricted versions of b -Firefighter and present an integer program for computing the exact value. We also discuss some interesting properties of the Expected Damage function.

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 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.

The decycling number $\nabla(G)$ of a graph G is the smallest number of vertices which can be removed from G so that the resultant graph contains no cycles. In this paper, we study the decycling number for the family of graphs consisting of the Cartesian product of two cycles. We completely solve the problem of determining the decycling number of $C_m \square C_n$ for all m and n. Moreover, we find a vertex set T that yields a maximum induced tree in $C_m\square C_n$.