Fire Containment in Planar Graphs

Esperet, Louis; Heuvel, Jan van den; Maffray, Frédéric; Sipma, Félix

doi:10.1002/jgt.21673

Cited by 21 publications

(15 citation statements)

References 3 publications

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“…Noting the results on the 2‐surviving rate for planar graphs without 3‐, 4‐, 5‐, or 6‐cycles, respectively , one may ask the following question: Problem Does it exist a function

f : N ∖ false{1, 2 false} \to {double-struckR}^{+}

such that for any

k \geq 3

and any nontrivial planar graph G without k ‐cycles there is

ρ_{2} false(G false) > f false(k false) ?

…”

Section: Remarksmentioning

confidence: 99%

“…It has been shown that 2 is the upper bound for triangle‐free planar graphs and planar graphs without 4‐, 5‐, or 6‐cycles , respectively. In particular Esperet et al.…”

Section: Introductionmentioning

confidence: 99%

“…In particular Esperet et al. have proven that

ρ_{2} false(G false) > \frac{1}{723636}

for any nontrivial triangle‐free planar graph, while Kong et al. that

ρ_{2} false(G false) > \frac{1}{76}

for any nontrivial planar graph without 4‐cycles.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the result implies a significant improvement of the bound for 2‐surviving rate for triangle‐free planar graphs (Esperet et al. ) and for planar graphs without 4‐cycles (Kong et al. ).…”

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confidence: 99%

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The 2‐surviving rate of planar graphs with average degree lower than

Gordinowicz

2018

Journal of Graph Theory

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Let G be any connected graph on n vertices, n≥2. Let k be any positive integer. Suppose that a fire breaks out at some vertex of G. Then, in each turn firefighters can protect at most k vertices of G not yet on fire; Next the fire spreads to all unprotected neighbors of burning vertices. The k‐surviving rate of G, denoted by ρkfalse(Gfalse), is the expected fraction of vertices that can be saved from the fire, provided that the starting vertex is chosen uniformly at random. In this note, it is shown that for any planar graph G with average degree 92−ε, where ε∈(0,1], there is ρ2false(Gfalse)≥29ε. In particular, the result implies a significant improvement of the bound for 2‐surviving rate for triangle‐free planar graphs (Esperet et al. ) and for planar graphs without 4‐cycles (Kong et al. ). The proof is done using the separator theorem for planar graphs.

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“…Noting the results on the 2‐surviving rate for planar graphs without 3‐, 4‐, 5‐, or 6‐cycles, respectively , one may ask the following question: Problem Does it exist a function

f : N ∖ false{1, 2 false} \to {double-struckR}^{+}

such that for any

k \geq 3

and any nontrivial planar graph G without k ‐cycles there is

ρ_{2} false(G false) > f false(k false) ?

…”

Section: Remarksmentioning

confidence: 99%

“…It has been shown that 2 is the upper bound for triangle‐free planar graphs and planar graphs without 4‐, 5‐, or 6‐cycles , respectively. In particular Esperet et al.…”

Section: Introductionmentioning

confidence: 99%

“…In particular Esperet et al. have proven that

ρ_{2} false(G false) > \frac{1}{723636}

for any nontrivial triangle‐free planar graph, while Kong et al. that

ρ_{2} false(G false) > \frac{1}{76}

for any nontrivial planar graph without 4‐cycles.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The 2‐surviving rate of planar graphs with average degree lower than

Gordinowicz

2018

Journal of Graph Theory

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“…The study of this concept has become very fruitful in the literature and the evidence is the existence of many works on the subject for different graph structures [9,6,14,16,29,36,41,44,42,43].…”

Section: Introductionmentioning

confidence: 99%

Firefighting as a Strategic Game

Àlvarez¹,

Blesa²,

Molter³

2015

Internet Mathematics

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The Firefighter Problem was proposed in 1995 [25] as a deterministic discrete-time model for the spread and containment of a fire. The problem is defined on an undirected finite graph G = (V, E), where initially fire breaks out at f nodes. In each subsequent time-step, two actions occur: A certain number b of firefighters are placed on nonburning nodes, permanently protecting them from the fire. Then the fire spreads to all non-defended neighbors of the nodes on fire. Since the graph is finite, at some point each node is either on fire or saved, and thus the fire cannot spread further. One of the objectives for the problem is to place the firefighters in such a way that the number of saved nodes is maximized.The applications of the Firefighter problem reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks. Most research on the problem considers the case in which the fire starts in a single place (i.e., f = 1), and in which the budget of available firefighters per time-step is one (i.e., b = 1). So does the work in this paper too. This configuration already leads to hard problems and even in this case the problem is known to be NP-hard. * This work was supported by grants TIN2013-46181-C2-1-R and TIN2012-37930-C02-2 of the Spanish Ministry of Science and Innovation, and by the project SGR 2014-1034 of the Generalitat de Catalunya.† A preliminary version of this work was presented in 11th Workshop on Algorithms and Models for the Web Graph (waw'14), Beijing, China, December 2014. 1In this work, we study the problem from a game-theoretical perspective. We introduce a strategic game model for the Firefighter Problem to tackle its complexity from a different angle. We refer to it as the Firefighter Game. Such a game-based context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination.At every time-step of the game, a player decides whether to place a new firefighter in a non-burning node of the graph. If so, he must decide where to place it. By placing it, the player is indirectly deciding which nodes to protect at that time-step. We define different utility functions in order to model selfish and non-selfish scenarios, which lead to equivalent games. We show that the Price of Anarchy (PoA) is linear for a particular family of graphs, but it is at most 2 for trees. We also analyze the quality of the equilibria when coalitions among players are allowed. It turns out that it is possible to compute an equilibrium in polynomial time, even for constant size coalitions. This yields to a polynomial time approximation algorithm for the problem and its approximation ratio equals the PoA of the corresponding game. We show that for some specific topologies, the PoA is constant when constant size coalitions are considered.

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