31,092 research outputs found
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Schnyder woods for higher genus triangulated surfaces
The final version of this extended abstract has been published in "Discrete and Computational Geometry (2009)"International audienceSchnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertex-spanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity
Spanning Tree Methods for Discriminative Training of Dependency Parsers
Untyped dependency parsing can be viewed as the problem of finding maximum spanning trees (MSTs) in directed graphs. Using this representation, the Eisner (1996) parsing algorithm is sufficient for searching the space of projective trees. More importantly, the representation is extended naturally to non-projective parsing using Chu-Liu-Edmonds (Chu and Liu, 1965; Edmonds, 1967) MST algorithm. These efficient parse search methods support large-margin discriminative training methods for learning dependency parsers. We evaluate these methods experimentally on the English and Czech treebanks
Energy, Laplacian energy of double graphs and new families of equienergetic graphs
For a graph with vertex set , the
extended double cover is a bipartite graph with bipartition (X, Y),
and , where two
vertices and are adjacent if and only if or adjacent to
in . The double graph of is a graph obtained by taking two
copies of and joining each vertex in one copy with the neighbours of
corresponding vertex in another copy. In this paper we study energy and
Laplacian energy of the graphs and , -spectra of the
-th iterated extended double cover of . We obtain a formula for the
number of spanning trees of . We also obtain some new families of
equienergetic and -equienergetic graphs.Comment: 23 pages, 1 figur
Exponents of Jacobians of Graphs and Regular Matroids
Let G be a finite undirected multigraph with no self-loops. The Jacobian Jac (G) is a finite abelian group associated with G whose cardinality is equal to the number of spanning trees of G. There are only a finite number of biconnected graphs G such that the exponent of Jac (G) equals 2 or 3. The definition of a Jacobian can also be extended to regular matroids as a generalization of graphs. We prove that there are finitely many connected regular matroids M such that Jac (M) has exponent 2 and characterize all such matroids
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