On rigidity, orientability and cores of random graphs with sliders

TL;DR

This paper investigates the emergence of giant rigid clusters in random graphs with sliders, revealing how the transition type depends on the fraction of freely moving points and introducing generalized concepts of cores and orientability.

Contribution

It introduces generalized notions of orientability and cores, determines thresholds for giant rigid components in graphs with sliders, and proves a conjecture about the 3+2-core size.

Findings

Giant rigid component threshold depends on fraction q of free points.

Transition is continuous for q ≤ 1/2 and discontinuous for q > 1/2.

Proves a conjecture about the size of the 3+2-core.

Abstract

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction $q$ of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph $G \in \mathcal{G}(n,c/n)$ the threshold value of $c$ for the appearance of a linear-sized rigid component as a function of $q$, generalizing results of Kasiviswanathan et al. We show that this appearance of a giant component undergoes a continuous transition for $q \leq 1/2$ and a discontinuous transition for $q > 1/2$. In our proofs, we introduce a generalized notion of orientability interpolating between 1- and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2+1-core and 3+2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan et al. about the size of the 3+2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.