Published on:

28 September 2023

Primary Category:

Combinatorics

Paper Authors:

Sean Dewar,

Georg Grasegger

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

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

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Coning relates cd and c*d by moving between dimensions

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

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Result attained by novel coning and approximation technique

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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