Efficient algorithms for the d-dimensional rigidity matroid of sparse graphs

  • Authors:

  • Sergey Bereg

  • Affiliations:

  • Department of Computer Science, University of Texas at Dallas, Box 830688, Richardson, TX 75083, USA

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2008

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Abstract

Let R"d(G) be the d-dimensional rigidity matroid for a graph G=(V,E). Combinatorial characterization of generically rigid graphs is known only for the plane d=2 [W. Whiteley, Rigidity and scene analysis, in: J.E. Goodman, J. O'Rourke (Eds.), Handbook of Discrete and Computational Geometry, CRC Press LLC, Boca Raton, FL, 2004, pp. 1327-1354, Chapter 60]. Recently Jackson and Jordan [B. Jackson, T. Jordan, The d-dimensional rigidity matroid of sparse graphs, Journal of Computational Theory (B) 95 (2005) 118-133] derived a min-max formula which determines the rank function in R"d(G) when G is sparse, i.e. has maximum degree at most d+2 and minimum degree at most d+1. We present efficient algorithms for sparse graphs G in higher dimensions d=3 that (i)detect if E is independent in the rigidity matroid for G, and (ii)construct G using vertex insertions preserving if G is isostatic, and (iii)compute the rank of R"d(G). The algorithms have linear running time assuming that the dimension d=3 is fixed.