Monte Carlo Results for Projected Self-Avoiding Polygons: A Two-dimensional Model for Knotted Polymers

We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of polymer passing on top of each other. Each polygon configuration can be viewed as the two- dimensional projection of a particular knot. We study numerically the statistics of large polygons with a fixed knot type, using a generalization of the BFACF algorithm for self-avoiding walks. This new algorithm incorporates both the displacement of crossings and the three types of Reidemeister transformations preserving the knot topology. Its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favor of this ergodicity are given together with a tentative sketch of proof. Assuming this ergodicity, we obtain numerically the following results for the statistics of knotted polygons: In the limit of a low crossing fugacity, we find a localization along the polygon of all the primary factors forming the knot. Increasing the crossing fugacity gives rise to a transition from a self-avoiding walk to a branched polymer behavior.Comment: 36 pages, 30 figures, latex, epsf. to appear in J.Phys.A: Math. Ge

Approximate min–max theorems for Steiner rooted-orientations of graphs and hypergraphs

Given an undirected hypergraph and a subset of vertices S subset of V with a specified root vertex r epsilon S, the STEINER ROOTFD-ORIENTATION problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the "connectivity" from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results of this paper are the following approximate min-max relations: Given an undirected hypergraph H, if S is 2k-hyperedge-connected in H, then H has a Steiner rooted k-hyperarc-connected orientation. Given an undirected graph G, if S is 2k-element-connected in G, then G has a Steiner rooted k-element-connected orientation. Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the STEINER TREE PACKING problem. Some complementary hardness results are presented at the end. (c) 2008 Elsevier Inc. All rights reserved

Enumerating kk-arc-connected orientations

12 pagesWe study the problem of enumerating the $k$-arc-connected orientations of a graph $G$, i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with delay $O(knm^2)$ and amortized time $O(m^2)$, which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the $\alpha$-orientations of a graph $G$ in delay $O(m^2)$ and for the outdegree sequences attained by $k$-arc-connected orientations of $G$ in delay $O(knm^2)$