83 research outputs found
Metric Dimension for Gabriel Unit Disk Graphs is NP-Complete
We show that finding a minimal number of landmark nodes for a unique virtual
addressing by hop-distances in wireless ad-hoc sensor networks is NP-complete
even if the networks are unit disk graphs that contain only Gabriel edges. This
problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The
Gabriel edges of a unit disc graph induce a planar O(\sqrt{n}) distance and an
optimal energy spanner. This is one of the most interesting restrictions of
Metric Dimension in the context of wireless multi-hop networks.Comment: A brief announcement of this result has been published in the
proceedings of ALGOSENSORS 201
Exact Localisations of Feedback Sets
The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform
a given multi digraph into an acyclic graph by deleting as few arcs
(vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one
of the classic NP-complete problems. An important contribution of this paper is
that the subgraphs , of all elementary
cycles or simple cycles running through some arc , can be computed in
and , respectively. We use
this fact and introduce the notion of the essential minor and isolated cycles,
which yield a priori problem size reductions and in the special case of so
called resolvable graphs an exact solution in . We show
that weighted versions of the FASP and FVSP possess a Bellman decomposition,
which yields exact solutions using a dynamic programming technique in times
and
, where , , respectively. The parameters can
be computed in , ,
respectively and denote the maximal dimension of the cycle space of all
appearing meta graphs, decoding the intersection behavior of the cycles.
Consequently, equal zero if all meta graphs are trees. Moreover, we
deliver several heuristics and discuss how to control their variation from the
optimum. Summarizing, the presented results allow us to suggest a strategy for
an implementation of a fast and accurate FASP/FVSP-SOLVER
On Metric Dimension of Functigraphs
The \emph{metric dimension} of a graph , denoted by , is the
minimum number of vertices such that each vertex is uniquely determined by its
distances to the chosen vertices. Let and be disjoint copies of a
graph and let be a function. Then a
\emph{functigraph} has the vertex set
and the edge set . We study how
metric dimension behaves in passing from to by first showing that
, if is a connected graph of order
and is any function. We further investigate the metric dimension of
functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure
On the metric dimension of a class of planar graphs
Let H = (V, E) be a non-trivial connected graph with vertex set V and edge set E. A set of ordered vertices Rm from V (H) is said to be a resolving set for H if each vertex of H is uniquely determined by its vector of distances to the vertices of Rm. The number of vertices in a smallest resolving set is called the metric dimension of H. In this article, we study the metric dimension for a rotationally symmetric family of planar graphs, each of which is shown to have an independent minimum resolving set of cardinality three.Publisher's Versio
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
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