Year: 2021
Type: preprint
Abstract: Consider a model of fire spreading through a graph; initially some vertices are burning, and at every given time-step fire spreads from burning vertices to their neighbours. The firefighter problem is a solitaire game in which a player is allowed, at every time-step, to protect some non-burning vertices (by effectively deleting them) in order to contain the fire growth. How many vertices per turn, on average, must be protected in order to stop the fire from spreading infinitely? Here we consider the problem on $\mathbb{Z}^2\times [h]$ for both nearest neighbour adjacency and strong adjacency. We determine the critical protection rates for these graphs to be $1.5h$ and $3h$, respectively. This establishes the fact that using an optimal two-dimensional strategy for all layers in parallel is asymptotically optimal.
Source: arXiv (Cornell University)
Authors: Deutch, Arye, Feldheim, Ohad Noy, Hod, Rani
Institutions: Hebrew University of Jerusalem, Tel Aviv University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4290692164