On the existence of paradoxical motions of generically rigid graphs on the sphere

08/01/2019
by   Matteo Gallet, et al.
0

We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with 3+3 vertices where no two vertices coincide or are antipodal.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 2

page 19

03/25/2020

On the Classification of Motions of Paradoxically Movable Graphs

Edge lengths of a graph are called flexible if there exist infinitely ma...
03/25/2019

A Reeb sphere theorem in graph theory

We prove a Reeb sphere theorem for finite simple graphs. The result brid...
03/06/2020

Reappraising the distribution of the number of edge crossings of graphs on a sphere

Many real transportation and mobility networks have their vertices place...
06/17/2018

Combinatorial manifolds are Hamiltonian

Extending a theorem of Whitney of 1931 we prove that all connected d-gra...
03/26/2020

FlexRiLoG – A SageMath Package for Motions of Graphs

In this paper we present the SageMath package FlexRiLoG (short for flexi...
01/16/2020

Extending drawings of complete graphs into arrangements of pseudocircles

Motivated by the successful application of geometry to proving the Harar...
01/17/2020

Globe-hopping

We consider versions of the grasshopper problem (Goulko and Kent, 2017) ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.