TY - JOUR

T1 - Multi-layered planar firefighting

AU - Deutsch, Arye

AU - Feldheim, Ohad Noy

AU - Hod, Rani

N1 - Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/12

Y1 - 2022/12

N2 - 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.

AB - 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.

KW - Firefighter problem

KW - Infinite graphs

KW - Solitaire game

UR - http://www.scopus.com/inward/record.url?scp=85135709645&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2022.113103

DO - 10.1016/j.disc.2022.113103

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AN - SCOPUS:85135709645

SN - 0012-365X

VL - 345

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 12

M1 - 113103

ER -