Constructing reflection-symmetric flexible realisations of graphs

TL;DR

This paper investigates reflection-symmetric graph realizations in the plane that can flex continuously while preserving edge lengths, identifying combinatorial conditions and constructing explicit symmetric realizations.

Contribution

It introduces a necessary combinatorial condition based on symmetric edge colorings for reflection-symmetric flexibility in graphs and provides methods to construct such realizations.

Findings

Identified a combinatorial condition for reflection-symmetric flexibility.

Constructed explicit symmetric realizations from certain colorings.

Analyzed a class of realizations composed of triangles and parallelograms.

Abstract

We study reflection-symmetric realisations of symmetric graphs in the plane that allow a continuous symmetry and edge-length preserving deformation. To do so, we identify a necessary combinatorial condition on graphs with reflection-symmetric flexible realisations. This condition is based on a specific type of edge colouring, where edges are assigned one of three colours in a symmetric way. From some of these colourings we also construct concrete reflection-symmetric realisations with their corresponding symmetry preserving motion. We study also a specific class of reflection-symmetric realisations consisting of triangles and parallelograms.