170 research outputs found
On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph
This paper studied the geometric and combinatorial aspects of the classical
Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega
be a height function which lifts the vertices of A into R3. Every flip in
triangulations of A can be associated with a direction. We first established a
relatively obvious relation between monotone sequences of directed flips
between triangulations of A and triangulations of the lifted point set of A in
R3. We then studied the structural properties of a directed flip graph (a
poset) on the set of all triangulations of A. We proved several general
properties of this poset which clearly explain when Lawson's algorithm works
and why it may fail in general. We further characterised the triangulations
which cause failure of Lawson's algorithm, and showed that they must contain
redundant interior vertices which are not removable by directed flips. A
special case if this result in 3d has been shown by B.Joe in 1989. As an
application, we described a simple algorithm to triangulate a special class of
3d non-convex polyhedra. We proved sufficient conditions for the termination of
this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure
On the Enumeration of Certain Weighted Graphs
We enumerate weighted graphs with a certain upper bound condition. We also
compute the generating function of the numbers of these graphs, and prove that
it is a rational function. In particular, we show that if the given graph is a
bipartite graph, then its generating function is of the form
, where is the number of vertices of the graph
and is a polynomial of degree at most .Comment: 25 page
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
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