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The number of equivalent realisations of rigid graphs

Published on:

28 September 2023

Primary Category:

Combinatorics

Paper Authors:

Sean Dewar,

Georg Grasegger

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Key Details

For rigid graphs, number of Euclidean realisations cd(G) can be defined

Likewise, number of spherical realisations c*d(G) can be defined

Coning relates cd and c*d by moving between dimensions

New proof shows cd(G) ≤ c*d(G) for any rigid graph G

Result attained by novel coning and approximation technique

AI generated summary

The number of equivalent realisations of rigid graphs

This paper proves that for any dimension d, the number of equivalent d-dimensional Euclidean realisations cd(G) of a rigid graph G is always less than or equal to the number of equivalent d-dimensional spherical realisations c*d(G). The key technique involves coning, adding a vertex adjacent to all others, which relates cd and c*d. Overall the work confirms cd(G) ≤ c*d(G) via coning and approximation arguments.

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