Embeddability and Stresses of Graphs

TL;DR

This paper explores the stress properties of various classes of graphs, establishing new bounds on stress freeness based on graph minors and embeddability, with implications for rigidity theory.

Contribution

It proves that linklessly embeddable graphs are generically 4-stress free and generalizes stress freeness results to K_{r+2}-minor free graphs for 0<r<5.

Findings

Linklessly embeddable graphs are generically 4-stress free

K_{r+2}-minor free graphs are generically r-stress free for 0<r<5

The results connect graph minors, embeddability, and stress properties

Abstract

Gluck (1975) has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that linklessly embeddable graphs are generically 4-stress free. Both of these results are corollaries of the following theorem: every K_{r+2}-minor free graph is generically r-stress free for 0<r<5. (This assertion is false for r>5.) We give an equivalent formulation of this theorem in the language of symmetric algebraic shifting and show that its analogue for exterior algebraic shifting also holds. Some further extensions are detailed.