113,462 research outputs found
On the threshold-width of graphs
The GG-width of a class of graphs GG is defined as follows. A graph G has
GG-width k if there are k independent sets N1,...,Nk in G such that G can be
embedded into a graph H in GG such that for every edge e in H which is not an
edge in G, there exists an i such that both endpoints of e are in Ni. For the
class TH of threshold graphs we show that TH-width is NP-complete and we
present fixed-parameter algorithms. We also show that for each k, graphs of
TH-width at most k are characterized by a finite collection of forbidden
induced subgraphs
-product and -threshold graphs
This paper is the continuation of the research of the author and his
colleagues of the {\it canonical} decomposition of graphs. The idea of the
canonical decomposition is to define the binary operation on the set of graphs
and to represent the graph under study as a product of prime elements with
respect to this operation. We consider the graph together with the arbitrary
partition of its vertex set into subsets (-partitioned graph). On the
set of -partitioned graphs distinguished up to isomorphism we consider the
binary algebraic operation (-product of graphs), determined by the
digraph . It is proved, that every operation defines the unique
factorization as a product of prime factors. We define -threshold graphs as
graphs, which could be represented as the product of one-vertex
factors, and the threshold-width of the graph as the minimum size of
such, that is -threshold. -threshold graphs generalize the classes of
threshold graphs and difference graphs and extend their properties. We show,
that the threshold-width is defined for all graphs, and give the
characterization of graphs with fixed threshold-width. We study in detail the
graphs with threshold-widths 1 and 2
Monotone properties of random geometric graphs have sharp thresholds
Random geometric graphs result from taking uniformly distributed points
in the unit cube, , and connecting two points if their Euclidean
distance is at most , for some prescribed . We show that monotone
properties for this class of graphs have sharp thresholds by reducing the
problem to bounding the bottleneck matching on two sets of points
distributed uniformly in . We present upper bounds on the threshold
width, and show that our bound is sharp for and at most a sublogarithmic
factor away for . Interestingly, the threshold width is much sharper for
random geometric graphs than for Bernoulli random graphs. Further, a random
geometric graph is shown to be a subgraph, with high probability, of another
independently drawn random geometric graph with a slightly larger radius; this
property is shown to have no analogue for Bernoulli random graphs.Comment: Published at http://dx.doi.org/10.1214/105051605000000575 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Maximizing Happiness in Graphs of Bounded Clique-Width
Clique-width is one of the most important parameters that describes
structural complexity of a graph. Probably, only treewidth is more studied
graph width parameter. In this paper we study how clique-width influences the
complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE)
problems. We answer a question of Choudhari and Reddy '18 about
parameterization by the distance to threshold graphs by showing that MHE is
NP-complete on threshold graphs. Hence, it is not even in XP when parameterized
by clique-width, since threshold graphs have clique-width at most two. As a
complement for this result we provide a algorithm for MHE, where is the number of colors
and is the clique-width of the input graph. We also
construct an FPT algorithm for MHV with running time
, where is the
number of colors in the input. Additionally, we show
algorithm for MHV on interval graphs.Comment: Accepted to LATIN 202
\order(\Gamma) Corrections to pair production in and collisions
Several schemes to introduce finite width effects to reactions involving
unstable elementary particles are given and the differences between them are
investigated. The effects of the different schemes is investigated numerically
for pair production. In we find that the effect of the
non-resonant graphs cannot be neglected for \sqrt{s}\geq400\GeV. There is no
difference between the various schemes to add these to the resonant graphs away
from threshold, although some violate gauge invariance. On the other hand, in
the reaction the effect of the non-resonant graphs is
large everywhere, due to the -channel pole. However, even requiring that the
outgoing lepton is observable () reduces the
contribution to about 1\%. Again, the scheme dependence is negligible here.Comment: 9 pages plus 6 with figures (.uu at end, also available with
anonymous ftp from pss058.psi.ch [129.129.40.58]), LaTeX, LMU-21/92,
PSI-PR-93-05, TTP92-3
Top quark production in e+e- annihilation
We analyze the four-fermion reactions e+e- -> 4f containing a single top
quark and three other fermions, a possible decay product of the resonant
anti-top quark, in the final state. This allows us to estimate the contribution
of the nonresonant Feynman graphs and effects related to the off mass shell
production and decay of the top quark. We test the sensitivity of the total
cross section at centre of mass energies in the t tbar threshold region and far
above it to the variation of the top quark width. We perform calculation in an
arbitrary linear gauge in the framework of the Standard Model and discuss an
important issue of gauge symmetry violation by the constant top quark width.Comment: 9 pages, 2 figures. A revised version accepted for publication in the
European Physical Journal C; a comment on the "fermion-loop scheme" modifie
- …