8,440 research outputs found
Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree
The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for
every tree there exists a natural number such that the following
holds: If is a -edge-connected simple graph with size divisible by
the size of , then can be edge-decomposed into subgraphs isomorphic to
. So far this conjecture has only been verified for paths, stars, and a
family of bistars. We prove a weaker version of the Tree Decomposition
Conjecture, where we require the subgraphs in the decomposition to be
isomorphic to graphs that can be obtained from by vertex-identifications.
We call such a subgraph a homomorphic copy of . This implies the Tree
Decomposition Conjecture under the additional constraint that the girth of
is greater than the diameter of . As an application, we verify the Tree
Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 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
Dimensional regularization of nonlinear sigma models on a finite time interval
We extend dimensional regularization to the case of compact spaces. Contrary
to previous regularization schemes employed for nonlinear sigma models on a
finite time interval (``quantum mechanical path integrals in curved space'')
dimensional regularization requires only a covariant finite two-loop
counterterm. This counterterm is nonvanishing and given by R/8.Comment: 9 pages, 7 figures, LaTeX, minor changes in text and reference
Generic Ising Trees
The Ising model on an infinite generic tree is defined as a thermodynamic
limit of finite systems. A detailed description of the corresponding
distribution of infinite spin configurations is given. As an application we
study the magnetization properties of such systems and prove that they exhibit
no spontaneous magnetization. Furthermore, the values of the Hausdorff and
spectral dimensions of the underlying trees are calculated and found to be,
respectively, and .Comment: 29 pages, 2 figures; typos corrected, one section and new references
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