DeepAI
Log In Sign Up

Multi-layered planar firefighting

05/08/2021
by   Arye Deutch, et al.
0

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 ℤ^2× [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.

READ FULL TEXT

page 1

page 2

page 3

page 4

03/09/2020

Adjacency Labelling for Planar Graphs (and Beyond)

We show that there exists an adjacency labelling scheme for planar graph...
07/30/2021

The Pyro game: a slow intelligent fire

In the Firefighter problem, a fire breaks out at a vertex of a graph and...
08/09/2019

Shorter Labeling Schemes for Planar Graphs

An adjacency labeling scheme for a given class of graphs is an algorithm...
12/21/2017

On Adjacency and e-adjacency in General Hypergraphs: Towards an e-adjacency Tensor

Adjacency between two vertices in graphs or hypergraphs is a pairwise re...
09/01/2018

On Adjacency and e-Adjacency in General Hypergraphs: Towards a New e-Adjacency Tensor

In graphs, the concept of adjacency is clearly defined: it is a pairwise...
07/20/2021

Non-existence of annular separators in geometric graphs

Benjamini and Papasoglou (2011) showed that planar graphs with uniform p...
04/09/2019

Discovering Bands from Graphs

Discovering the underlying structure of a given graph is one of the fund...