Fire Containment in Planar Graphs

TL;DR

This paper investigates the minimum number of firefighters needed to ensure a positive expected number of vertices are saved from fire in planar graphs, establishing bounds for general and triangle-free cases.

Contribution

It proves that at most 4 firefighters are needed for planar graphs and exactly 2 for triangle-free planar graphs to achieve a positive survival rate.

Findings

At most 4 firefighters suffice for planar graphs.

Exactly 2 firefighters suffice for triangle-free planar graphs.

Ensures linearly many vertices are saved in these graph classes.

Abstract

In a graph $G$, a fire starts at some vertex. At every time step, firefighters can protect up to $k$ vertices, and then the fire spreads to all unprotected neighbours. The $k$-surviving rate $\rho_k(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs $\cG$ we are interested in the minimum value $k$ such that $\rho_k(G)\ge\epsilon$ for some constant $\epsilon>0$ and all $G\in\cG$ i.e., such that linearly many vertices are expected to be saved in every graph from $\cG$). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.