Flexible placements of periodic graphs in the plane

TL;DR

This paper develops combinatorial methods to determine when periodic graphs can be flexibly embedded in the plane, considering fixed and flexible lattices of periodicity, advancing understanding of graph flexibility in geometric and periodic contexts.

Contribution

It introduces NBAC-colourings for quotient gain graphs to identify flexible periodic embeddings with fixed or flexible lattices, providing new characterizations for 1- and 2-periodic graphs.

Findings

Identifies conditions for flexible embeddings with fixed lattice.

Characterizes flexible embeddings with flexible lattice for 1-periodic graphs.

Provides special case characterizations for 2-periodic graphs.

Abstract

Given a periodic graph, we wish to determine via combinatorial methods whether it has periodic embeddings in the plane that -- via motions that preserve edge-lengths and periodicity -- can be continuously deformed into another non-congruent embedding of the graph. By introducing NBAC-colourings for the corresponding quotient gain graphs, we identify which periodic graphs have flexible embeddings in the plane when the lattice of periodicity is fixed. We further characterise with NBAC-colourings which 1-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity, and characterise in special cases which 2-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity.