43 research outputs found
A Note on Flips in Diagonal Rectangulations
Rectangulations are partitions of a square into axis-aligned rectangles. A
number of results provide bijections between combinatorial equivalence classes
of rectangulations and families of pattern-avoiding permutations. Other results
deal with local changes involving a single edge of a rectangulation, referred
to as flips, edge rotations, or edge pivoting. Such operations induce a graph
on equivalence classes of rectangulations, related to so-called flip graphs on
triangulations and other families of geometric partitions. In this note, we
consider a family of flip operations on the equivalence classes of diagonal
rectangulations, and their interpretation as transpositions in the associated
Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This
complements results from Law and Reading (JCTA, 2012) and provides a complete
characterization of flip operations on diagonal rectangulations, in both
geometric and combinatorial terms
Enumerating -arc-connected orientations
12 pagesWe study the problem of enumerating the -arc-connected orientations of a graph , i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with delay and amortized time , which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the -orientations of a graph in delay and for the outdegree sequences attained by -arc-connected orientations of in delay
Geometric biplane graphs I: maximal graphs
Postprint (author’s final draft
Geometric biplane graphs I: maximal graphs
We study biplane graphs drawn on a finite planar point set in general position. This is the family of geometric graphs whose vertex set is and can be decomposed into two plane graphs. We show that two maximal biplane graphs-in the sense that no edge can be added while staying biplane-may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over -element point sets.Peer ReviewedPostprint (author's final draft
Computational design of planar multistable compliant structures
This paper presents a method for designing planar multistable compliant structures. Given a sequence of desired stable states and the corresponding poses of the structure, we identify the topology and geometric realization of a mechanism—consisting of bars and joints—that is able to physically reproduce the desired multistable behavior. In order to solve this problem efficiently, we build on insights from minimally rigid graph theory to identify simple but effective topologies for the mechanism. We then optimize its geometric parameters, such as joint positions and bar lengths, to obtain correct transitions between the given poses. Simultaneously, we ensure adequate stability of each pose based on an effective approximate error metric related to the elastic energy Hessian of the bars in the mechanism. As demonstrated by our results, we obtain functional multistable mechanisms of manageable complexity that can be fabricated using 3D printing. Further, we evaluated the effectiveness of our method on a large number of examples in the simulation and fabricated several physical prototypes
Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
In this work we consider triangulations of point sets in the Euclidean plane,
i.e., maximal straight-line crossing-free graphs on a finite set of points.
Given a triangulation of a point set, an edge flip is the operation of removing
one edge and adding another one, such that the resulting graph is again a
triangulation. Flips are a major way of locally transforming triangular meshes.
We show that, given a point set in the Euclidean plane and two
triangulations and of , it is an APX-hard problem to minimize
the number of edge flips to transform to .Comment: A previous version only showed NP-completeness of the corresponding
decision problem. The current version is the one of the accepted manuscrip