1,030 research outputs found
Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop
We consider quadrangulations with a boundary and derive explicit expressions
for the generating functions of these maps with either a marked vertex at a
prescribed distance from the boundary, or two boundary vertices at a prescribed
mutual distance in the map. For large maps, this yields explicit formulas for
the bulk-boundary and boundary-boundary correlators in the various encountered
scaling regimes: a small boundary, a dense boundary and a critical boundary
regime. The critical boundary regime is characterized by a one-parameter family
of scaling functions interpolating between the Brownian map and the Brownian
Continuum Random Tree. We discuss the cases of both generic and self-avoiding
boundaries, which are shown to share the same universal scaling limit. We
finally address the question of the bulk-loop distance statistics in the
context of planar quadrangulations equipped with a self-avoiding loop. Here
again, a new family of scaling functions describing critical loops is
discovered.Comment: 55 pages, 14 figures, final version with minor correction
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure
Recognizing Partial Cubes in Quadratic Time
We show how to test whether a graph with n vertices and m edges is a partial
cube, and if so how to find a distance-preserving embedding of the graph into a
hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time
solutions.Comment: 25 pages, five figures. This version significantly expands previous
versions, including a new report on an implementation of the algorithm and
experiments with i
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