198 research outputs found
Enumeration of m-ary cacti
The purpose of this paper is to enumerate various classes of cyclically
colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is
motivated by the topological classification of complex polynomials having at
most m critical values, studied by Zvonkin and others. We obtain explicit
formulae for both labelled and unlabelled m-ary cacti, according to i) the
number of polygons, ii) the vertex-color distribution, iii) the vertex-degree
distribution of each color. We also enumerate m-ary cacti according to the
order of their automorphism group. Using a generalization of Otter's formula,
we express the species of m-ary cacti in terms of rooted and of pointed cacti.
A variant of the m-dimensional Lagrange inversion is then used to enumerate
these structures. The method of Liskovets for the enumeration of unrooted
planar maps can also be adapted to m-ary cacti.Comment: LaTeX2e, 28 pages, 9 figures (eps), 3 table
Arc Operads and Arc Algebras
Several topological and homological operads based on families of projectively
weighted arcs in bounded surfaces are introduced and studied. The spaces
underlying the basic operad are identified with open subsets of a
compactification due to Penner of a space closely related to Riemann's moduli
space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras,
where the entire BV structure is realized simplicially. Furthermore, our basic
operad contains the cacti operad up to homotopy, and it similarly acts on the
loop space of any topological space. New operad structures on the circle are
classified and combined with the basic operad to produce geometrically natural
extensions of the algebraic structure of BV algebras, which are also computed.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper15.abs.htm
Grassmann-Berezin Calculus and Theorems of the Matrix-Tree Type
We prove two generalizations of the matrix-tree theorem. The first one, a
result essentially due to Moon for which we provide a new proof, extends the
``all minors'' matrix-tree theorem to the ``massive'' case where no condition
on row or column sums is imposed. The second generalization, which is new,
extends the recently discovered Pfaffian-tree theorem of Masbaum and Vaintrob
into a ``Hyperpfaffian-cactus'' theorem. Our methods are noninductive, explicit
and make critical use of Grassmann-Berezin calculus that was developed for the
needs of modern theoretical physics.Comment: 23 pages, 2 figures, 3 references adde
Generating partitions of a graph into a fixed number of minimum weight cuts
AbstractIn this paper, we present an algorithm for the generation of all partitions of a graph G with positive edge weights into k mincuts. The algorithm is an enumeration procedure based on the cactus representation of the mincuts of G. We report computational results demonstrating the efficiency of the algorithm in practice and describe in more detail a specific application for generating cuts in branch-and-cut algorithms for the traveling salesman problem
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