Rigid partitions: from high connectivity to random graphs

TL;DR

This paper introduces new conditions for graph rigidity based on rigid partitions, extending previous work, and applies these to analyze the rigidity properties of various classes of random and dense graphs.

Contribution

It provides novel sufficient conditions for graph rigidity using rigid partitions and applies these to establish rigidity results for random, highly-connected, and dense graphs.

Findings

Random $C d ext{log} d$-regular graphs are typically $d$-rigid

Existence of a giant $d$-rigid component in sparse random binomial graphs

Rigidity of sparse bipartite graphs similar to complete bipartite graphs

Abstract

A graph is called $d$-rigid if there exists a generic embedding of its vertex set into $\mathbb{R}^d$ such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal $d$ such that the graph is $d$-rigid. We present new sufficient conditions for the $d$-rigidity of a graph in terms of the existence of ``rigid partitions'' -- partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz. As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random $C d\log d$-regular graphs are typically $d$-rigid, demonstrate the existence of a giant $d$-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting $\binom{d+1}{2}$ disjoint connected dominating sets is $d$-rigid. This implies a weak version of the Lov\'asz--Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for $d$-rigidity in the random graph process typically coincides with the hitting time for minimum degree $d$.