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International Workshop on Automated Deduction in Geometry

ADG 2014: Automated Deduction in Geometry pp 129–148Cite as

On Flattenability of Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9201))

Abstract

We consider a generalization of the concept of d-flattenability of graphs - introduced for the \(l_2\) norm by Belk and Connelly - to general \(l_p\) norms, with integer P, \(1 \le p < \infty \), though many of our results work for \(l_\infty \) as well. The following results are shown for graphs G, using notions of genericity, rigidity, and generic d-dimensional rigidity matroid introduced by Kitson for frameworks in general \(l_p\) norms, as well as the cones of vectors of pairwise \(l_p^p\) distances of a finite point configuration in d-dimensional, \(l_p\) space: (i) d-flattenability of a graph G is equivalent to the convexity of d-dimensional, inherent Cayley configurations spaces for G, a concept introduced by the first author; (ii) d-flattenability and convexity of Cayley configuration spaces over specified non-edges of a d-dimensional framework are not generic properties of frameworks (in arbitrary dimension); (iii) d-flattenability of G is equivalent to all of G’s generic frameworks being d-flattenable; (iv) existence of one generic d-flattenable framework for G is equivalent to the independence of the edges of G, a generic property of frameworks; (v) the rank of G equals the dimension of the projection of the d-dimensional stratum of the \(l_p^p\) distance cone. We give stronger results for specific norms for \(d = 2\): we show that (vi) 2-flattenable graphs for the \(l_1\)-norm (and \(l_\infty \)-norm) are a larger class than 2-flattenable graphs for Euclidean \(l_2\)-norm case and finally (vii) prove further results towards characterizing 2-flattenability in the \(l_1\)-norm. A number of conjectures and open problems are posed.

M. Sitharam—This research was supported in part by the grant NSF CCF-1117695

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Acknowledgement

We thank Bob Connelly, Steven Gortler and Derek Kitson for interesting conversations related to this paper.

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Authors and Affiliations

  1. University of Florida, Gainesville, Florida

    Meera Sitharam & Joel Willoughby

Authors
  1. Meera Sitharam
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  2. Joel Willoughby
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Correspondence to Joel Willoughby .

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Editors and Affiliations

  1. University of Vigo, Pontevedra, Spain

    Francisco Botana

  2. University of Coimbra, Coimbra, Portugal

    Pedro Quaresma

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Sitharam, M., Willoughby, J. (2015). On Flattenability of Graphs. In: Botana, F., Quaresma, P. (eds) Automated Deduction in Geometry. ADG 2014. Lecture Notes in Computer Science(), vol 9201. Springer, Cham. https://doi.org/10.1007/978-3-319-21362-0_9

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  • DOI: https://doi.org/10.1007/978-3-319-21362-0_9

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  • Online ISBN: 978-3-319-21362-0

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