Firefighting on square, hexagonal, and triangular grids

TL;DR

This paper analyzes the firefighter problem on various grid types, providing bounds and strategies for fire containment and vertex saving in finite and infinite cases, with specific results for square and hexagonal grids.

Contribution

It offers new bounds on surviving rates for finite grids, defines and computes rates for infinite grids, and introduces strategies and conjectures for fire containment.

Findings

Surviving rate for finite square grid is approximately 5/8.

Infinite square grid has a surviving rate of 1/4.

Additional protection can significantly slow fire spread.

Abstract

In this paper, we consider the \emph{firefighter problem} on a graph $G=(V,E)$ that is either finite or infinite. Suppose that a fire breaks out at a given vertex $v \in V$. In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible (if $G$ is finite) or to stop the fire from spreading (for an infinite case). The surviving rate $\rho(G)$ of a finite graph $G$ is defined as the expected percentage of vertices that can be saved when a fire breaks out at a vertex of $G$ that is selected uniformly random. For a finite square grid $P_n \square P_n$, we show that $5/8 + o(1) \le \rho(P_n \square P_n) \le 67243/105300 + o(1)$ (leaving the gap smaller than 0.014) and conjecture that the surviving rate is asymptotic to 5/8. We define the surviving rate for infinite graphs and prove it to be 1/4 for the infinite square grid, even in the case of finitely many initial fires. For the infinite hexagonal grid we provide a winning strategy if two additional vertices can be protected at any point of the process, and we conjecture that the firefighter has no strategy to stop the fire without additional help. We also show how the speed of the spreading fire can be reduced by a constant factor.