1,188 research outputs found
Distributive Lattices, Polyhedra, and Generalized Flow
A D-polyhedron is a polyhedron such that if are in then so are
their componentwise max and min. In other words, the point set of a
D-polyhedron forms a distributive lattice with the dominance order. We provide
a full characterization of the bounding hyperplanes of D-polyhedra.
Aside from being a nice combination of geometric and order theoretic
concepts, D-polyhedra are a unifying generalization of several distributive
lattices which arise from graphs. In fact every D-polyhedron corresponds to a
directed graph with arc-parameters, such that every point in the polyhedron
corresponds to a vertex potential on the graph. Alternatively, an edge-based
description of the point set can be given. The objects in this model are dual
to generalized flows, i.e., dual to flows with gains and losses.
These models can be specialized to yield some cases of distributive lattices
that have been studied previously. Particular specializations are: lattices of
flows of planar digraphs (Khuller, Naor and Klein), of -orientations of
planar graphs (Felsner), of c-orientations (Propp) and of -bonds of
digraphs (Felsner and Knauer). As an additional application we exhibit a
distributive lattice structure on generalized flow of breakeven planar
digraphs.Comment: 17 pages, 3 figure
Digraphs and cycle polynomials for free-by-cyclic groups
Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be
represented by an expanding, irreducible train-track map. The automorphism
determines a free-by-cyclic group
and a homomorphism . By work of Neumann,
Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, has an open cone
neighborhood in whose integral points
correspond to other fibrations of whose associated outer automorphisms
are themselves representable by expanding irreducible train-track maps. In this
paper, we define an analog of McMullen's Teichm\"uller polynomial that computes
the dilatations of all outer automorphism in .Comment: 41 pages, 20 figure
Primitive digraphs with large exponents and slowly synchronizing automata
We present several infinite series of synchronizing automata for which the
minimum length of reset words is close to the square of the number of states.
All these automata are tightly related to primitive digraphs with large
exponent.Comment: 23 pages, 11 figures, 3 tables. This is a translation (with a
slightly updated bibliography) of the authors' paper published in Russian in:
Zapiski Nauchnyh Seminarov POMI [Kombinatorika i Teorija Grafov. IV], Vol.
402, 9-39 (2012), see ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v402/p009.pdf
Version 2: a few typos are correcte
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