133 research outputs found
A Survey on Alliances and Related Parameters in Graphs
In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, -domination, -independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global -alliances in graphs.We also propose a new framework, called (global) -alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations
Alliance free and alliance cover sets
A \emph{defensive} (\emph{offensive}) -\emph{alliance} in
is a set such that every in (in the boundary of ) has
at least more neighbors in than it has in . A set
is \emph{defensive} (\emph{offensive}) -\emph{alliance free,}
if for all defensive (offensive) -alliance , ,
i.e., does not contain any defensive (offensive) -alliance as a subset.
A set is a \emph{defensive} (\emph{offensive})
-\emph{alliance cover}, if for all defensive (offensive) -alliance ,
, i.e., contains at least one vertex from each
defensive (offensive) -alliance of . In this paper we show several
mathematical properties of defensive (offensive) -alliance free sets and
defensive (offensive) -alliance cover sets, including tight bounds on the
cardinality of defensive (offensive) -alliance free (cover) sets
Nova Geminorum 1912 and the Origin of the Idea of Gravitational Lensing
Einstein's early calculations of gravitational lensing, contained in a
scratch notebook and dated to the spring of 1912, are reexamined. A hitherto
unknown letter by Einstein suggests that he entertained the idea of explaining
the phenomenon of new stars by gravitational lensing in the fall of 1915 much
more seriously than was previously assumed. A reexamination of the relevant
calculations by Einstein shows that, indeed, at least some of them most likely
date from early October 1915. But in support of earlier historical
interpretation of Einstein's notes, it is argued that the appearance of Nova
Geminorum 1912 (DN Gem) in March 1912 may, in fact, provide a relevant context
and motivation for Einstein's lensing calculations on the occasion of his first
meeting with Erwin Freundlich during a visit in Berlin in April 1912. We also
comment on the significance of Einstein's consideration of gravitational
lensing in the fall of 1915 for the reconstruction of Einstein's final steps in
his path towards general relativity.Comment: 31 p
Fast branching algorithm for Cluster Vertex Deletion
In the family of clustering problems, we are given a set of objects (vertices
of the graph), together with some observed pairwise similarities (edges). The
goal is to identify clusters of similar objects by slightly modifying the graph
to obtain a cluster graph (disjoint union of cliques). Hueffner et al. [Theory
Comput. Syst. 2010] initiated the parameterized study of Cluster Vertex
Deletion, where the allowed modification is vertex deletion, and presented an
elegant O(2^k * k^9 + n * m)-time fixed-parameter algorithm, parameterized by
the solution size. In our work, we pick up this line of research and present an
O(1.9102^k * (n + m))-time branching algorithm
Towards Optimal and Expressive Kernelization for d-Hitting Set
d-Hitting Set is the NP-hard problem of selecting at most k vertices of a
hypergraph so that each hyperedge, all of which have cardinality at most d,
contains at least one selected vertex. The applications of d-Hitting Set are,
for example, fault diagnosis, automatic program verification, and the
noise-minimizing assignment of frequencies to radio transmitters.
We show a linear-time algorithm that transforms an instance of d-Hitting Set
into an equivalent instance comprising at most O(k^d) hyperedges and vertices.
In terms of parameterized complexity, this is a problem kernel. Our
kernelization algorithm is based on speeding up the well-known approach of
finding and shrinking sunflowers in hypergraphs, which yields problem kernels
with structural properties that we condense into the concept of expressive
kernelization.
We conduct experiments to show that our kernelization algorithm can kernelize
instances with more than 10^7 hyperedges in less than five minutes.
Finally, we show that the number of vertices in the problem kernel can be
further reduced to O(k^{d-1}) with additional O(k^{1.5 d}) processing time by
nontrivially combining the sunflower technique with d-Hitting Set problem
kernels due to Abu-Khzam and Moser.Comment: This version gives corrected experimental results, adds additional
figures, and more formally defines "expressive kernelization
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