Symmetric contact systems of segments, pseudotriangulations and inductive constructions for corresponding surface graphs

TL;DR

This paper characterizes surface graphs derived from symmetric contact systems of segments and pseudotriangulations, extending known results and introducing an inductive method with implications for geometric rigidity.

Contribution

It provides a new inductive characterization of surface graphs from symmetric contact systems and pseudotriangulations, generalizing previous theorems and exploring rigidity implications.

Findings

Characterization of quotient surface graphs from symmetric contact systems

Extension of Thomassen's and Streinu-Haas results to symmetric cases

Introduction of an inductive construction method for these surface graphs

Abstract

We characterise the quotient surface graphs arising from symmetric contact systems of line segments in the plane and also from symmetric pointed pseudotriangulations in the case where the group of symmetries is generated by a translation or a rotation of finite order. These results generalise well known results of Thomassen, in the case of line segments, and of Streinu and Haas et al., in the case of pseudotriangulations. Our main tool is a new inductive characterisation of the appropriate classes of surface graphs. We also discuss some consequences of our results in the area of geometric rigidity theory.