Cycle decompositions and train tracks

Braid Forcing and Star-shaped Train Tracks

Global results are proved about the way in which Boyland’s forcing partial order organizes a set of braid types: those of periodic orbits of Smale’s horseshoe map for which the associated train track is a star. This is a special case of a conjecture introduced in [dCHb], which claims that forcing organizes all horseshoe braid types into linearly ordered families which are, in turn, parameterize...

5-cycle Decompositions from Paired 3- and 4-cycle Decompositions

Let (V, T ) be a 3-fold triple system and (V,C) a 4-fold 4-cycle system on the same set V . This choice of indices 3 and 4 ensures that each system contains the same number of cycles: |T | = |C|. We pair up the cycles, {t, c}, where t ∈ T and c ∈ C, in such a way that t and c share one edge. If t = (x, y, z) and c = (x, y, u, v), so t and c share the edge {x, y}, then we retain the 5-cycle (z, ...

Path And Cycle Decompositions

First I want to say a few words about my graph terminology. If I want to allow loops, I use the adjective reflexive. If I want to allow multiple edges, I use multigraph. Thus, a graph has no loops and no multiple edges. I use valency rather than degree. If we say a graph is 4-valent (or tetravalent), it means it is regular of valency 4, for example. A decomposition of a graph X is a partition o...

Planar Decompositions and Cycle Constraints

Dual-decomposition methods for optimization have emerged as an extremely powerful tool for solving combinatorial problems in graphical models. These techniques can be thought of as decomposing a complex model into a collection of easier-to-solve components, providing a variational bound which can then be optimized over its parameters. A wide variety of algorithms have been proposed, often disti...

Trend-Cycle Decompositions