Coincident Rigidity of 2-Dimensional Frameworks

TL;DR

This paper explores the conditions under which 2D frameworks with coincident vertices are rigid, extending existing characterizations to larger sets of coincident vertices and verifying the conjecture for triplets.

Contribution

It formulates a conjecture generalizing previous results on coincident rigidity and confirms it for the case of three coincident vertices.

Findings

Characterization of 2D infinitesimally rigid frameworks with coincident vertices

Verification of the conjecture for three vertices in the set T

Extension of rigidity theory to arbitrary sets of coincident vertices

Abstract

Fekete, Jord\'an and Kaszanitzky [4] characterised the graphs which can be realised as 2-dimensional, infinitesimally rigid, bar-joint frameworks in which two given vertices are coincident. We formulate a conjecture which would extend their characterisation to an arbitrary set T of vertices and verify our conjecture when |T| = 3.