Generic combinatorial rigidity of periodic frameworks

TL;DR

This paper provides a combinatorial characterization of minimal rigidity in planar periodic frameworks, analogous to the Maxwell-Laman Theorem, enabling polynomial-time verification using combinatorial algorithms.

Contribution

It introduces a new combinatorial criterion for rigidity of periodic frameworks and develops related concepts like periodic direction networks and Z2-graded-sparse colored graphs.

Findings

Characterization of minimal rigidity for planar periodic frameworks

Development of polynomial-time algorithms for rigidity verification

Introduction of periodic direction networks and Z2-graded-sparse colored graphs

Abstract

We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z2-graded-sparse colored graphs.