Mike Develin, Stephen G. Hartke
We consider a deterministic discrete-time model of fire spread introduced by Hartnell [1995] and the problem of minimizing the number of burnt vertices when deploying a limited number of firefighters per timestep. We consider the process occurring on the d-dimensional square lattice for d>=3, and we prove several results, including two conjectures of Wang and Moeller [2002].
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In a graph $G$, a fire starts at some vertex. At every time step, firefighters can protect up to $k$ vertices, and then the fire spreads to all unprotected neighbours. The $k$-surviving rate $ρ_k(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs $\cG$ we are interested in the minimum value $k$ such that $ρ_k(G)\geε$ for some constant $ε>0$ and all $G\in\cG$ i.e., such that linearly many vertices are expected to be saved in every graph from $\cG$). In this not…
In this article, we study Hartnell's Firefighter Problem through the group theoretic notions of growth and quasi-isometry. A graph has the $n$-containment property if for every finite initial fire, there is a strategy to contain the fire by protecting $n$ vertices at each turn. A graph has the constant containment property if there is an integer $n$ such that it has the $n$-containment property. Our first result is that any locally finite connected graph with quadratic growth has the constant containment property; the converse does not hold. This result provides a unified way to recover previo…
The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell. The game on a graph $G$ can be described as follows: let $f_n$ be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time $n\geq 1$, $f_n$ vertices which are not on fire become protected, and then the fire spreads to all unprotected neighbors of vertices on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The graph $G$ has the \emph{$f_n$-containment property} if every initial fire admits an strategy that prote…
The Firefighter Problem (FP) is a graph problem originally introduced in 1995 to model the spread of a fire in a graph, which has attracted considerable attention in the literature. The goal is to devise a strategy to employ a given sequence of firefighters on strategic points in the graph in order to contain efficiently the fire (which spreads from each unprotected vertex to all of it neighbours on successive time steps). Recently, an online version of FP---where the number of firefighters available at each turn are revealed in real-time--- has been introduced in the literature and studied on…
In this paper, we consider the \emph{firefighter problem} on a graph $G=(V,E)$ that is either finite or infinite. Suppose that a fire breaks out at a given vertex $v \in V$. In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible (if $G$ is finite) or to stop the fire from spreading (for an infinite case). The surviving rate $ρ(G)$ of a finite graph $G$ is defined as the expected percentage of vertices that ca…
The firefighter problem with $k$ firefighters on an infinite graph $G$ is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex $v\in V(G)$ on Turn 1. On each subsequent even turn, $k$ firefighters protect $k$ vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters' goal is to eventually stop the spread of the fire. If there exists a strategy for $k$ firefighters to eventually stop the spread of the fire, then we say $G$ is $k$-containable. We conside…
A monotone grid class is a permutation class (i.e., a downset of permutations under the containment order) defined by local monotonicity conditions. We give a simplified proof of a result of Murphy and Vatter that monotone grid classes of forests are partially well-ordered.
We study subclasses of grid intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we provide a comprehensive study of classes of graphs between grid intersection graphs and bipartite permutation graphs and the containment relation on these classes. Order dimension plays a role in many arguments.
The Infectious Vaccination Problem: a variant of Firefighting with Spreading Defence
研究消防员问题变体，其中防御可扩散，提供算法、难度及无限网格图的遏制结果。
The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid…
In the Firefighter problem, a fire breaks out at a vertex of a graph and at each subsequent time step, the firefighter chooses a vertex to protect and then the fire spreads from each burned vertex to every unprotected neighbour. The problem can be thought of as a simplified model for the spread of gossip or disease in a network. We introduce a new two-player variation called the Pyro game, in which at each step, the fire spreads from one burned vertex to all unprotected neighbours of that vertex. The fire is no longer automated and aims to maximize the number of burned vertices. We show, that …
COVID on trees and infinite grids
基于Hartnell的firefighting模型，提出Unburning Algorithm和Containment Protocol，分别解决有根树和一般图上的病毒传播问题，证明算法性能并验证整数间隙猜想。
Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The burning number of a graph $G$ is the minimum number of rounds necessary for each vertex of $G$ to burn. We consider the burning number of the $m \times n$ Cartesian grid graphs, written $G_{m,n}$.\ For $m = ω(\sqrt{n})$, the asymptotic value of the burning number of $G_{m,n}$ was determined, but only the growth rate …
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