1,738 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
Limit theory of combinatorial optimization for random geometric graphs
In the random geometric graph , vertices are placed randomly in
Euclidean -space and edges are added between any pair of vertices distant at
most from each other. We establish strong laws of large numbers (LLNs)
for a large class of graph parameters, evaluated for in the
thermodynamic limit with const., and also in the dense limit with , . Examples include domination number,
independence number, clique-covering number, eternal domination number and
triangle packing number. The general theory is based on certain subadditivity
and superadditivity properties, and also yields LLNs for other functionals such
as the minimum weight for the travelling salesman, spanning tree, matching,
bipartite matching and bipartite travelling salesman problems, for a general
class of weight functions with at most polynomial growth of order
, under thermodynamic scaling of the distance parameter.Comment: 64 page
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
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
Eternal dominating sets on digraphs and orientations of graphs
We study the eternal dominating number and the m-eternal dominating number on
digraphs. We generalize known results on graphs to digraphs. We also consider
the problem "oriented (m-)eternal domination", consisting in finding an
orientation of a graph that minimizes its eternal dominating number. We prove
that computing the oriented eternal dominating number is NP-hard and
characterize the graphs for which the oriented m-eternal dominating number is
2. We also study these two parameters on trees, cycles, complete graphs,
complete bipartite graphs, trivially perfect graphs and different kinds of
grids and products of graphs.Comment: 34 page
Network-centric methods for heterogeneous multiagent systems
We present tools for a network topology based characterization of heterogeneity in multiagent systems, thereby providing a framework for the analysis and design of heterogeneous multiagent networks from a network structure view-point. In heterogeneous networks, agents with a diverse set of resources coordinate with each other. Coordination among different
agents and the structure of the underlying network topology have significant impacts on the overall behavior and functionality of the system. Using constructs from graph theory, a qualitative as well as a quantitative analysis is performed to examine an inter-relationship between the network topology and the distribution of agents with various capabilities in heterogeneous networks. Our goal is to allow agents maximally exploit heterogeneous resources available within the network through local interactions, thus exploring a promise heterogeneous networks hold to accomplish complicated tasks by leveraging upon the assorted capabilities of agents. For a reliable operations of such systems, the issue of security against intrusions and malicious agents is also addressed. We provide a scheme to secure a network against a sequence of intruder attacks through a set of heterogeneous guards. Moreover, robustness of networked systems against noise corruption and structural changes in the underlying network topology is also examined.Ph.D
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