The firefighter problem on polynomial and intermediate growth groups

Amir, Gideon; Baldasso, Rangel; Kozma, Gady

doi:10.1016/j.disc.2020.112077

Cited by 7 publications

(3 citation statements)

References 3 publications

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“…Proof. This is proved exactly as in [1] (see the proof of Theorem 1, and, in particular, Equation (4) there). We skip the details.…”

Section: Proof Of Theorem 13supporting

confidence: 58%

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Fire retainment on Cayley graphs

Amir¹,

Baldasso²,

Герасимова³

et al. 2021

Preprint

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We study the fire-retaining problem on groups, a quasiisometry invariant 1 introduced by Martínez-Pedroza and Prytu la [8], related to the firefighter problem. We prove that any Cayley graph with degree-d polynomial growth does not satisfy {f (n)}-retainment, for any f (n) = o(n d−2 ), matching the upper bound given for the firefighter problem for these graphs. In the exponential growth regime we prove general lower bounds for direct products and wreath products. These bounds are tight, and show that for exponential-growth groups a wide variety of behaviors is possible. In particular, we construct, for any d ≥ 1, groups that satisfy {n d }-retainment but not o(n d )-retainment, as well as groups that do not satisfy sub-exponential retainment.

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“…Proof. This is proved exactly as in [1] (see the proof of Theorem 1, and, in particular, Equation (4) there). We skip the details.…”

Section: Proof Of Theorem 13supporting

confidence: 58%

“…Develin and Hartke [2] conjectured the converse should hold for the ddimensional integer lattice, and this was recently verified to hold for Cayley graphs of groups with polynomial growth by Amir, Baldasso, and Kozma [1].…”

Section: Introductionmentioning

confidence: 86%

Fire retainment on Cayley graphs

Amir¹,

Baldasso²,

Герасимова³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Develin and Hartke [6] generalized Fogarty's argument to show that 2d − 1 firefighters are required to contain a single-source fire in Z d and conjectured that for f(t) t d−2 , there exists an initial fire such that any strategy using f(t) firefighters fails to contain it. A new result by Amir, Baldasso and Kozma [2] use a Fogarty-type argument together with an isoperimetric tool to prove a generalized version of Develin and Hartke's conjecture, namely that in Cayley graphs with polynomial growth, any strategy with f(t) t d−2 cannot contain a large enough initial fire.…”

Section: Introductionmentioning

confidence: 99%

Multi-layered planar firefighting

Deutch¹,

Feldheim²,

Hod³

2021

Preprint

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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 neighbors. 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 neighbor 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.

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Coarse geometry of the fire retaining property and group splittings

Martínez-Pedroza

Prytuła

2023

Geom Dedicata

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The firefighter problem on polynomial and intermediate growth groups

Cited by 7 publications

References 3 publications

Fire retainment on Cayley graphs

Fire retainment on Cayley graphs

Multi-layered planar firefighting

Coarse geometry of the fire retaining property and group splittings

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