160 research outputs found
Connectivity and tree structure in finite graphs
Considering systems of separations in a graph that separate every pair of a
given set of vertex sets that are themselves not separated by these
separations, we determine conditions under which such a separation system
contains a nested subsystem that still separates those sets and is invariant
under the automorphisms of the graph.
As an application, we show that the -blocks -- the maximal vertex sets
that cannot be separated by at most vertices -- of a graph live in
distinct parts of a suitable tree-decomposition of of adhesion at most ,
whose decomposition tree is invariant under the automorphisms of . This
extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a
similar theorem of Tutte for .
Under mild additional assumptions, which are necessary, our decompositions
can be combined into one overall tree-decomposition that distinguishes, for all
simultaneously, all the -blocks of a finite graph.Comment: 31 page
A short proof that every finite graph has a tree-decomposition displaying its tangles
This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.ejc.2016.04.007We give a short proof that every finite graph (or matroid) has a tree-decomposition that displays all maximal tangles.
This theorem for graphs is a central result of the graph minors project of Robertson and Seymour and the extension to matroids is due to Geelen, Gerards and Whittle.Emmanuel Colleg
Infinite Graphic Matroids
An infinite matroid is graphic if all of its finite minors are graphic
and the intersection of any circuit with any cocircuit is finite. We
show that a matroid is graphic if and only if it can be represented by a
graph-like topological space: that is, a graph-like space in the sense of
Thomassen and Vella. This extends Tutte’s characterization of finite
graphic matroids.
Working in the representing space, we prove that any circuit in a
3-connected graphic matroid is countable
A Liouville hyperbolic souvlaki
We construct a transient bounded-degree graph no transient subgraph of which embeds in any surface of finite genus. Moreover, we construct a transient, Liouville, bounded-degree, Gromov– hyperbolic graph with trivial hyperbolic boundary that has no transient subtree. This answers a question of Benjamini. This graph also yields a (further) counterexample to a conjecture of Benjamini and Schramm. In an appendix by G´abor Pete and Gourab Ray, our construction is extended to yield a unimodular graph with the above properties
Topological cycle matroids of infinite graphs
We prove that the topological cycles of an arbitrary infinite graph together with its topological ends form a matroid. This matroid is, in general, neither finitary nor cofinitary.Emmanuel Colleg
Anomalous Dynamics of Translocation
We study the dynamics of the passage of a polymer through a membrane pore
(translocation), focusing on the scaling properties with the number of monomers
. The natural coordinate for translocation is the number of monomers on one
side of the hole at a given time. Commonly used models which assume Brownian
dynamics for this variable predict a mean (unforced) passage time that
scales as , even in the presence of an entropic barrier. However, the time
it takes for a free polymer to diffuse a distance of the order of its radius by
Rouse dynamics scales with an exponent larger than 2, and this should provide a
lower bound to the translocation time. To resolve this discrepancy, we perform
numerical simulations with Rouse dynamics for both phantom (in space dimensions
and 2), and self-avoiding (in ) chains. The results indicate that
for large , translocation times scale in the same manner as diffusion times,
but with a larger prefactor that depends on the size of the hole. Such scaling
implies anomalous dynamics for the translocation process. In particular, the
fluctuations in the monomer number at the hole are predicted to be
non-diffusive at short times, while the average pulling velocity of the polymer
in the presence of a chemical potential difference is predicted to depend on
.Comment: 9 pages, 9 figures. Submitted to Physical Review
Single chain structure in thin polymer films: Corrections to Flory's and Silberberg's hypotheses
Conformational properties of polymer melts confined between two hard
structureless walls are investigated by Monte Carlo simulation of the
bond-fluctuation model. Parallel and perpendicular components of chain
extension, bond-bond correlation function and structure factor are computed and
compared with recent theoretical approaches attempting to go beyond Flory's and
Silberberg's hypotheses. We demonstrate that for ultrathin films where the
thickness, , is smaller than the excluded volume screening length (blob
size), , the chain size parallel to the walls diverges logarithmically,
with . The corresponding bond-bond
correlation function decreases like a power law, with
being the curvilinear distance between bonds and . % Upon increasing
the film thickness, , we find -- in contrast to Flory's hypothesis -- the
bulk exponent and, more importantly, an {\em decreasing}
that gives direct evidence for an {\em enhanced} self-interaction of chain
segments reflected at the walls. Systematic deviations from the Kratky plateau
as a function of are found for the single chain form factor parallel to the
walls in agreement with the {\em non-monotonous} behaviour predicted by theory.
This structure in the Kratky plateau might give rise to an erroneous estimation
of the chain extension from scattering experiments. For large the
deviations are linear with the wave vector, , but are very weak. In
contrast, for ultrathin films, , very strong corrections are found
(albeit logarithmic in ) suggesting a possible experimental verification of
our results.Comment: 16 pages, 7 figures. Dedicated to L. Sch\"afer on the occasion of his
60th birthda
Distance dependence of angular correlations in dense polymer solutions
Angular correlations in dense solutions and melts of flexible polymer chains
are investigated with respect to the distance between the bonds by
comparing quantitative predictions of perturbation calculations with numerical
data obtained by Monte Carlo simulation of the bond-fluctuation model. We
consider both monodisperse systems and grand-canonical (Flory-distributed)
equilibrium polymers. Density effects are discussed as well as finite chain
length corrections. The intrachain bond-bond correlation function is
shown to decay as for \xi \ll r \ll \r^* with being
the screening length of the density fluctuations and a novel
length scale increasing slowly with (mean) chain length .Comment: 17 pages, 5 figures, accepted for publication at Macromolecule
Small-Angle Excess Scattering: Glassy Freezing or Local Orientational Ordering?
We present Monte Carlo simulations of a dense polymer melt which shows
glass-transition-like slowing-down upon cooling, as well as a build up of
nematic order. At small wave vectors q this model system shows excess
scattering similar to that recently reported for light-scattering experiments
on some polymeric and molecular glass-forming liquids. For our model system we
can provide clear evidence that this excess scattering is due to the onset of
short-range nematic order and not directly related to the glass transition.Comment: 3 Pages of Latex + 4 Figure
Monte Carlo simulations of random copolymers at a selective interface
We investigate numerically using the bond--fluctuation model the adsorption
of a random AB--copolymer at the interface between two solvents. From our
results we infer several scaling relations: the radius of gyration of the
copolymer in the direction perpendicular to the interface () scales
with , the interfacial selectivity strength, as
where is the usual Flory exponent and
is the copolymer's length; furthermore the monomer density at the interface
scales as for small . We also determine numerically the
monomer densities in the two solvents and discuss their dependence on the
distance from the interface.Comment: Latex text file appended with figures.tar.g
- …