963 research outputs found
New results on metric-locating-dominating sets of graphs
A dominating set of a graph is a metric-locating-dominating set if each
vertex of the graph is uniquely distinguished by its distances from the
elements of , and the minimum cardinality of such a set is called the
metric-location-domination number. In this paper, we undertake a study that, in
general graphs and specific families, relates metric-locating-dominating sets
to other special sets: resolving sets, dominating sets, locating-dominating
sets and doubly resolving sets. We first characterize classes of trees
according to certain relationships between their metric-location-domination
number and their metric dimension and domination number. Then, we show
different methods to transform metric-locating-dominating sets into
locating-dominating sets and doubly resolving sets. Our methods produce new
bounds on the minimum cardinalities of all those sets, some of them involving
parameters that have not been related so far.Comment: 13 pages, 3 figure
Metric Dimension of Amalgamation of Graphs
A set of vertices resolves a graph if every vertex is uniquely
determined by its vector of distances to the vertices in . The metric
dimension of is the minimum cardinality of a resolving set of .
Let be a finite collection of graphs and each
has a fixed vertex or a fixed edge called a terminal
vertex or edge, respectively. The \emph{vertex-amalgamation} of , denoted by , is formed by taking all
the 's and identifying their terminal vertices. Similarly, the
\emph{edge-amalgamation} of , denoted by
, is formed by taking all the 's and identifying
their terminal edges.
Here we study the metric dimensions of vertex-amalgamation and
edge-amalgamation for finite collection of arbitrary graphs. We give lower and
upper bounds for the dimensions, show that the bounds are tight, and construct
infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on
Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised
version 21 December 201
Metric-locating-dominating sets of graphs for constructing related subsets of vertices
© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of S , and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them concerning parameters that have not been related so farPeer ReviewedPostprint (author's final draft
Riemannian Optimization for Convex and Non-Convex Signal Processing and Machine Learning Applications
The performance of most algorithms for signal processing and machine learning applications highly depends on the underlying optimization algorithms. Multiple techniques have been proposed for solving convex and non-convex problems such as interior-point methods and semidefinite programming. However, it is well known that these algorithms are not ideally suited for large-scale optimization with a high number of variables and/or constraints. This thesis exploits a novel optimization method, known as Riemannian optimization, for efficiently solving convex and non-convex problems with signal processing and machine learning applications. Unlike most optimization techniques whose complexities increase with the number of constraints, Riemannian methods smartly exploit the structure of the search space, a.k.a., the set of feasible solutions, to reduce the embedded dimension and efficiently solve optimization problems in a reasonable time. However, such efficiency comes at the expense of universality as the geometry of each manifold needs to be investigated individually. This thesis explains the steps of designing first and second-order Riemannian optimization methods for smooth matrix manifolds through the study and design of optimization algorithms for various applications. In particular, the paper is interested in contemporary applications in signal processing and machine learning, such as community detection, graph-based clustering, phase retrieval, and indoor and outdoor location determination. Simulation results are provided to attest to the efficiency of the proposed methods against popular generic and specialized solvers for each of the above applications
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