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Which graphs are rigid in ℓ p d ?

Sean Dewar1, Derek Kitson2,3, Anthony Nixon2

  • 1Johann Radon Institute of Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria.

Journal of Global Optimization : an International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering
|May 9, 2022
PubMed
Summary

Graphs are minimally rigid in d-dimensional Lp-space if and only if they are (d, d)-tight. This study introduces a graph bracing operation and proves independence for sparse graphs and triangulations in Lp-space.

Keywords:
Bar-joint frameworkInfinitesimal rigidityNormed spacesRigidity matroid

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Area of Science:

  • Graph theory
  • Geometric rigidity
  • Metric spaces

Background:

  • The study of graph rigidity is crucial in understanding the stability of structures.
  • Minimal rigidity in Euclidean space is well-understood, but less so in general Lp-spaces.
  • The conjecture links (d, d)-tightness to minimal rigidity in d-dimensional Lp-space.

Purpose of the Study:

  • To investigate the conjecture that a graph is minimally rigid in d-dimensional Lp-space if and only if it is (d, d)-tight.
  • To extend the understanding of graph rigidity beyond Euclidean spaces to more general Lp-spaces.
  • To provide new tools and proofs supporting this conjecture.

Main Methods:

  • Introduction of a graph bracing operation that preserves independence in rigidity matroids.
  • Proof of independence for (d, d)-sparse graphs within specific degree bounds in Lp-space.
  • Demonstration of minimal rigidity for triangulations of the projective plane in Lp^3-space.
  • Development of rigidity-preserving graph moves for strictly convex and smooth normed spaces.
  • Showing independence of triangulations of the sphere in 3D spaces within this class.

Main Results:

  • A graph bracing operation is presented that maintains independence when transitioning between Lp-spaces of different dimensions.
  • Graphs that are (d, d)-sparse, with minimum degree at most d+1 and maximum degree at most d+2, are proven to be independent in Lp^d-space.
  • Triangulations of the projective plane are shown to be minimally rigid in Lp^3-space.
  • Rigidity-preserving graph moves are cataloged for a broader class of normed spaces.
  • Triangulations of the sphere are demonstrated to be independent in 3D spaces within these normed spaces.

Conclusions:

  • The results provide strong support for the conjecture relating minimal rigidity and (d, d)-tightness in Lp-spaces.
  • New methods and findings advance the understanding of graph rigidity in non-Euclidean metric spaces.
  • The study contributes to the theory of combinatorial rigidity and its applications.