371 research outputs found
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
Computing the Geometric Intersection Number of Curves
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most l on a combinatorial surface of complexity n we describe simple algorithms to (1) compute the geometric intersection number of c in O(n+ l^2) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O(n+l^4) time, (3) decide if the geometric intersection number of c is zero, i.e. if c is homotopic to a simple curve, in O(n+l log^2 l) time.
To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O(n+g^2l^2) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2). Interestingly, our solution to problem (3) is the first quasi-linear algorithm since the problem was raised by Poincare more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most l in O(n+ l^2) time
A Geometrically Based Approach to 3D Skeleton Curve Blending
Projet SYNTIMThis paper presents an efficient method to smoothly transform one polyline into another. The method makes use of a moving adapted frame associated to each curve. % Several choices are presented and discussed for the moving frame. We introduce a simple propagation equation involving the moving frame which accounts for the transition between two consecutive points of a polyline. The curve transformation is performed using the interpolation of the quantities involved in this propagation equation. Depending on the type of interpolation, we propose two algorithms to compute the intermediary shape of the curve. Moreover, we define a measure of the transformation and minimize this measure to distinguish a «minimal transformation» which the user can modify through the addition of simple parameters
Algorithms for Length Spectra of Combinatorial Tori
Consider a weighted, undirected graph cellularly embedded on a topological
surface. The function assigning to each free homotopy class of closed curves
the length of a shortest cycle within this homotopy class is called the marked
length spectrum. The (unmarked) length spectrum is obtained by just listing the
length values of the marked length spectrum in increasing order.
In this paper, we describe algorithms for computing the (un)marked length
spectra of graphs embedded on the torus. More specifically, we preprocess a
weighted graph of complexity in time so that, given a
cycle with edges representing a free homotopy class, the length of a
shortest homotopic cycle can be computed in time. Moreover,
given any positive integer , the first values of its unmarked length
spectrum can be computed in time .
Our algorithms are based on a correspondence between weighted graphs on the
torus and polyhedral norms. In particular, we give a weight independent bound
on the complexity of the unit ball of such norms. As an immediate consequence
we can decide if two embedded weighted graphs have the same marked spectrum in
polynomial time. We also consider the problem of comparing the unmarked spectra
and provide a polynomial time algorithm in the unweighted case and a randomized
polynomial time algorithm otherwise.Comment: 33 pages, 16 figure
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