87 research outputs found
A survey of graph burning
Graph burning is a deterministic, discrete-time process that models how
influence or contagion spreads in a graph. Associated to each graph is its
burning number, which is a parameter that quantifies how quickly the influence
spreads. We survey results on graph burning, focusing on bounds, conjectures,
and algorithms related to the burning number. We will discuss state-of-the-art
results on the burning number conjecture, burning numbers of graph classes, and
algorithmic complexity. We include a list of conjectures, variants, and open
problems on graph burning
A survey of graph burning
Graph burning is a deterministic, discrete-time process that models how influence or contagion spreads in a graph. Associated to each graph is its burning number, which is a parameter that quantifies how quickly the influence spreads. We survey results on graph burning, focusing on bounds, conjectures, and algorithms related to the burning number. We will discuss state-of-the-art results on the burning number conjecture, burning numbers of graph classes, and algorithmic complexity. We include a list of conjectures, variants, and open problems on graph burning
Parameterized Complexity of Graph Burning
Graph Burning asks, given a graph and an integer , whether
there exists such that every vertex in
has distance at most from some . This problem is known to be
NP-complete even on connected caterpillars of maximum degree . We study the
parameterized complexity of this problem and answer all questions arose by Kare
and Reddy [IWOCA 2019] about parameterized complexity of the problem. We show
that the problem is W[2]-complete parameterized by and that it does no
admit a polynomial kernel parameterized by vertex cover number unless
. We also show that the problem is
fixed-parameter tractable parameterized by clique-width plus the maximum
diameter among all connected components. This implies the fixed-parameter
tractability parameterized by modular-width, by treedepth, and by distance to
cographs. Although the parameterization by distance to split graphs cannot be
handled with the clique-width argument, we show that this is also tractable by
a reduction to a generalized problem with a smaller solution size.Comment: 10 pages, 2 figures, IPEC 202
On the approximability of the burning number
The burning number of a graph is the smallest number such that the
vertices of can be covered by balls of radii . As
computing the burning number of a graph is known to be NP-hard, even on trees,
it is natural to consider polynomial time approximation algorithms for the
quantity. The best known approximation factor in the literature is for
general graphs and for trees. In this note we give a
-approximation algorithm for the burning
number of general graphs, and a PTAS for the burning number of trees and
forests. Moreover, we show that computing a
-approximation of the burning number of a general graph
is NP-hard.Comment: 7 pages, no figures. Comments are welcome
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Spread and defend infection in graphs
The spread of an infection, a contagion, meme, emotion, message and various
other spreadable objects have been discussed in several works. Burning and
firefighting have been discussed in particular on static graphs. Graph burning
simulates the notion of the spread of "fire" throughout a graph (plus, one
unburned node burned at each time-step); graph firefighting simulates the
defending of nodes by placing firefighters on the nodes which have not been
already burned while the fire is being spread (started by only a single fire
source).
This article studies a combination of firefighting and burning on a graph
class which is a variation (generalization) of temporal graphs. Nodes can be
infected from "outside" a network. We present a notion of both upgrading (of
unburned nodes, similar to firefighting) and repairing (of infected nodes). The
nodes which are burned, firefighted, or repaired are chosen probabilistically.
So a variable amount of nodes are allowed to be infected, upgraded and repaired
in each time step.
In the model presented in this article, both burning and firefighting proceed
concurrently, we introduce such a system to enable the community to study the
notion of spread of an infection and the notion of upgrade/repair against each
other. The graph class that we study (on which, these processes are simulated)
is a variation of temporal graph class in which at each time-step,
probabilistically, a communication takes place (iff an edge exists in that time
step). In addition, a node can be "worn out" and thus can be removed from the
network, and a new healthy node can be added to the network as well. This class
of graphs enables systems with high complexity to be able to be simulated and
studied
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