On generic 3-rigidity of graphs

TL;DR

This paper presents a necessary condition for the generic 3-rigidity of graphs based on edge partitioning, highlighting that the condition is not sufficient for guaranteeing 3-rigidity.

Contribution

It introduces a new necessary condition for generic 3-rigidity using edge partitioning into subsets that form 2-rigid graphs, advancing understanding of graph rigidity.

Findings

The condition is necessary but not sufficient for 3-rigidity.

Partitioning edges into subsets can reveal rigidity properties.

The work builds on prior concepts of 2-rigidity and edge-deletion effects.

Abstract

We give a necessary condition of generic 3 -rigidity of graphs relying on partitioning the edges into 3 subsets; such that each subset-pair gives a generically 2-rigid graph, either by themselves or after an appropriate edge-deletion. Notably, as pointed out by Dewar and Gallet, the condition is still not sufficient.