On Flattenability of Graphs

• Published in: Automated Deduction in Geometry Vol. 9201; pp. 129 - 148
• Main Authors: Sitharam, Meera, Willoughby, Joel
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2015; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Arbitrary Dimension; Artificial intelligence; Configuration Space; Distance Vector; Euclidean Distance Matrix; General Norm; Graphics programming; Mathematical theory of computation
• ISBN: 331921361X; 9783319213613
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-21362-0_9

We consider a generalization of the concept of d-flattenability of graphs - introduced for the l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_2$$\end{document} norm by Belk and Connelly - to general lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_p$$\end{document} norms, with integer P, 1≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le p < \infty $$\end{document}, though many of our results work for l∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_\infty $$\end{document} 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 lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_p$$\end{document} norms, as well as the cones of vectors of pairwise lpp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_p^p$$\end{document} distances of a finite point configuration in d-dimensional, lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_p$$\end{document} 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 lpp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_p^p$$\end{document} distance cone. We give stronger results for specific norms for d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d = 2$$\end{document}: we show that (vi) 2-flattenable graphs for the l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_1$$\end{document}-norm (and l∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_\infty $$\end{document}-norm) are a larger class than 2-flattenable graphs for Euclidean l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_2$$\end{document}-norm case and finally (vii) prove further results towards characterizing 2-flattenability in the l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_1$$\end{document}-norm. A number of conjectures and open problems are posed.