160 research outputs found

    Connectivity and tree structure in finite graphs

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    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 kk-blocks -- the maximal vertex sets that cannot be separated by at most kk vertices -- of a graph GG live in distinct parts of a suitable tree-decomposition of GG of adhesion at most kk, whose decomposition tree is invariant under the automorphisms of GG. This extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a similar theorem of Tutte for k=2k=2. Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all kk simultaneously, all the kk-blocks of a finite graph.Comment: 31 page

    A short proof that every finite graph has a tree-decomposition displaying its tangles

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    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

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    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

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    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

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    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

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    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 NN. 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 τ\tau that scales as N2N^2, 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 d=1d=1 and 2), and self-avoiding (in d=2d=2) chains. The results indicate that for large NN, 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 NN.Comment: 9 pages, 9 figures. Submitted to Physical Review

    Small-Angle Excess Scattering: Glassy Freezing or Local Orientational Ordering?

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    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

    Single chain structure in thin polymer films: Corrections to Flory's and Silberberg's hypotheses

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    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, HH, is smaller than the excluded volume screening length (blob size), ξ\xi, the chain size parallel to the walls diverges logarithmically, R2/2Nb2+clog(N)R^2/2N \approx b^2 + c \log(N) with c1/Hc \sim 1/H. The corresponding bond-bond correlation function decreases like a power law, C(s)=d/sωC(s) = d/s^{\omega} with ss being the curvilinear distance between bonds and ω=1\omega=1. % Upon increasing the film thickness, HH, we find -- in contrast to Flory's hypothesis -- the bulk exponent ω=3/2\omega=3/2 and, more importantly, an {\em decreasing} d(H)d(H) 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 HH 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 HH the deviations are linear with the wave vector, qq, but are very weak. In contrast, for ultrathin films, H<ξH<\xi, very strong corrections are found (albeit logarithmic in qq) 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

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    Angular correlations in dense solutions and melts of flexible polymer chains are investigated with respect to the distance rr 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 P(r)P(r) is shown to decay as P(r)1/r3P(r) \sim 1/r^3 for \xi \ll r \ll \r^* with ξ\xi being the screening length of the density fluctuations and rN1/3r^* \sim N^{1/3} a novel length scale increasing slowly with (mean) chain length NN.Comment: 17 pages, 5 figures, accepted for publication at Macromolecule

    Monte Carlo simulations of random copolymers at a selective interface

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    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 (RgzR_{gz}) scales with χ\chi, the interfacial selectivity strength, as Rgz=Nνf(Nχ)R_{gz}=N^{\nu}f(\sqrt{N}\chi) where ν\nu is the usual Flory exponent and NN is the copolymer's length; furthermore the monomer density at the interface scales as χ2ν\chi^{2\nu} for small χ\chi. 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
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