5,299 research outputs found
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Guarding Networks Through Heterogeneous Mobile Guards
In this article, the issue of guarding multi-agent systems against a sequence
of intruder attacks through mobile heterogeneous guards (guards with different
ranges) is discussed. The article makes use of graph theoretic abstractions of
such systems in which agents are the nodes of a graph and edges represent
interconnections between agents. Guards represent specialized mobile agents on
specific nodes with capabilities to successfully detect and respond to an
attack within their guarding range. Using this abstraction, the article
addresses the problem in the context of eternal security problem in graphs.
Eternal security refers to securing all the nodes in a graph against an
infinite sequence of intruder attacks by a certain minimum number of guards.
This paper makes use of heterogeneous guards and addresses all the components
of the eternal security problem including the number of guards, their
deployment and movement strategies. In the proposed solution, a graph is
decomposed into clusters and a guard with appropriate range is then assigned to
each cluster. These guards ensure that all nodes within their corresponding
cluster are being protected at all times, thereby achieving the eternal
security in the graph.Comment: American Control Conference, Chicago, IL, 201
An Eternal Domination Problem in Grids
A dynamic domination problem in graphs is considered in which an infinite sequence of attacks occur at vertices with mobile guards; the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard. Other guards are allowed to move at the same time, and before and after each attack, the vertices containing guards must form a dominating set of the graph. The minimum number of guards that can defend the graph against such an arbitrary sequence of attacks is called the m-eviction number of the graph. In this paper, the m-eviction number is determined exactly for grids with and upper bounds are given for all
Eternal Independent Sets in Graphs
The use of mobile guards to protect a graph has received much attention in the literature of late in the form of eternal dominating sets, eternal vertex covers and other models of graph protection. In this paper, eternal independent sets are introduced. These are independent sets such that the following can be iterated forever: a vertex in the independent set can be replaced with a neighboring vertex and the resulting set is independent
Vertex covers and eternal dominating sets
AbstractThe eternal domination problem requires a graph to be protected against an infinitely long sequence of attacks on vertices by guards located at vertices, the configuration of guards inducing a dominating set at all times. An attack at a vertex with no guard is defended by sending a guard from a neighboring vertex to the attacked vertex. We allow any number of guards to move to neighboring vertices at the same time in response to an attack. We compare the eternal domination number with the vertex cover number of a graph. One of our main results is that the eternal domination number is less than the vertex cover number of any graph of minimum degree at least two having girth at least nine
Closing the Gap: Eternal Domination on 3 x n Grids
The domination number for grid graphs has been a long studied problem; the first results appeared over thirty years ago [Jacobson 1984] and the final results appeared in 2013 [Goncalves 2013]. Grid graphs are a natural class of graphs to consider for the eternal dominating set problem as the domination number forms a lower bound for the eternal domination number. Â The 3 x n grid has been considered in several papers, and the difference between the upper and lower bounds for the eternal domination number in the all-guards move model has been reduced to a linear function of n. In this short paper, we provide an upper bound for the eternal domination number which exceeds the lower bound by at most 3
Holographic probabilities in eternal inflation
In the global description of eternal inflation, probabilities for vacua are
notoriously ambiguous. The local point of view is preferred by holography and
naturally picks out a simple probability measure. It is insensitive to large
expansion factors or lifetimes, and so resolves a recently noted paradox. Any
cosmological measure must be complemented with the probability for observers to
emerge in a given vacuum. In lieu of anthropic criteria, I propose to estimate
this by the entropy that can be produced in a local patch. This allows for
prior-free predictions.Comment: 5 pages, 3 figures. v4: published version, misprints corrected (mu ->
eta
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