Multi-layered planar firefighting

Arye Deutsch, Ohad Noy Feldheim, Rani Hod*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 Z2×[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 languageAmerican English
Article number113103
Pages (from-to)1-18
Number of pages18
JournalDiscrete Mathematics
Volume345
Issue number12
DOIs
StatePublished - 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

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