Infinitesimal rigidity in normed planes

Sean Dewar

doi:10.1137/19M1284051

Infinitesimal rigidity in normed planes

Sean Dewar

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

8 Citations (Scopus)

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Abstract

We prove that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a $(2,2)$-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph $K_4$ by considering smoothness and strict convexity properties of the unit ball.

Original language	English
Pages (from-to)	1205 - 1231
Number of pages	27
Journal	SIAM Journal on Discrete Mathematics
Volume	34
Issue number	2
DOIs	https://doi.org/10.1137/19M1284051
Publication status	Published - 26 May 2020

Bibliographical note

27 pages

Keywords

math.MG
52C25, 52A21

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This is the accepted author manuscript (AAM). The final published version (version of record) is available online via SIAM at https://doi.org/10.1137/19M1284051. Please refer to any applicable terms of use of the publisher.
Accepted author manuscript, 350 KBLicence: CC BY

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AB - We prove that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a $(2,2)$-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph $K_4$ by considering smoothness and strict convexity properties of the unit ball.

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Infinitesimal rigidity in normed planes

Abstract

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