## 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 Z^{2}×[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.

Original language | American English |
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Article number | 113103 |

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 345 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2022 |

### Bibliographical note

Funding Information:Research supported by Len Blavatnik and the Blavatnik Family foundation .

Publisher Copyright:

© 2022 Elsevier B.V.

## Keywords

- Firefighter problem
- Infinite graphs
- Solitaire game