相关论文: The Coarse Geometry of Hartnell's Firefighter Prob…
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…
We study the fire-retaining problem on groups, a quasi-isometry invariant introduced by Mart\'inez-Pedroza and Prytula [8], related to the firefighter problem. We prove that any Cayley graph with degree-$d$ polynomial growth does not…
Given a non-decreasing function $f \colon \mathbb{N} \to \mathbb{N}$ we define a single player game on (infinite) connected graphs that we call fire retaining. If a graph $G$ admits a winning strategy for any initial configuration (initial…
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…
We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not satisfy $\{f(n)\}$-containment for any $f=o(n^{d-2})$. This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that…
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…
The Firefighting problem is defined as follows. At time $t=0$, a fire breaks out at a vertex of a graph. At each time step $t \geq 0$, a firefighter permanently defends (protects) an unburned vertex, and the fire then spread to all…
The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and…
We investigate a new oriented variant of the Firefighter Problem. In the traditional Firefighter Problem, a fire breaks out at a given vertex of a graph, and at each time interval spreads to neighbouring vertices that have not been…
We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be…
The severity of wildfires can be mitigated adopting preventive measures like the construction of firebreaks that are strips of land from which the vegetation is completely removed. In this paper, we model the problem of wildfire containment…
The firefighter problem is defined as below. A fire initially breaks out at a vertex r on a graph G. In each step, a firefighter chooses to protect one vertex, which is not yet burnt. And the fire spreads out to its unprotected neighboring…
Let $G$ be any connected graph on $n$ vertices, $n \ge 2.$ Let $k$ be any positive integer. Suppose that a fire breaks out on some vertex of $G.$ Then in each turn $k$ firefighters can protect vertices of $G$ --- each can protect one vertex…
We use Hartnell's model for virus spread on a graph, also known as firefighting. For rooted trees, we propose an Unburning Algorithm, a type of greedy algorithm starting from the leaves and working back towards the root. We show that the…
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 $\rho_k(G)$ of $G$ is the expectation of the…
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…
In the Firefighter problem, introduced by Hartnell in 1995, a fire spreads through a graph while a player chooses which vertices to protect in order to contain it. In this paper, we focus on the case of trees and we consider as well the…
It is well known that fighting a fire is a hard task. The Firefighter problem asks how to optimally deploy firefighters to defend the vertices of a graph from a fire. This problem is NP-Complete on all but a few classes of graphs.…
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…
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…