Abstract
A direction–length framework is a pair (G,p) where G=(V;D,L) is a ‘mixed’ graph whose edges are labelled as ‘direction’ or ‘length’ edges and p is a map from V to ℝd for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G + be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G,p) is bounded if and only if (G +,p) is infinitesimally rigid. This gives a characterization of when (G,p) is bounded in terms of the rank of the rigidity matrix of (G +,p). We use this to characterize when a mixed graph is generically bounded in ℝd. As an application we deduce that if (G,p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G∖e,p) is bounded.
Article PDF
Similar content being viewed by others
References
Berg, A., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proceedings of the 11th Annual European Symposium on Algorithms 2003. Springer Lecture Notes in Computer Science, vol. 2832, pp. 78–89 (2003)
Connelly, R.: A flexible sphere. Math. Intell. 1, 130–131 (1978)
Connelly, R.: Generic global rigidity. Discrete Comput. Geom. 33, 549–563 (2005)
Euler, L.: Opera Postuma, vol. 1. Petropoli (1862), pp. 494–496. Euler Archive index number E819, at http://math.dartmouth.edu/~euler/
Gortler, S., Healy, A., Thurston, D.: Characterizing generic global rigidity, at arXiv:0710.0926 (2007)
Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21, 65–84 (1992)
Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory, Ser. B 94, 1–29 (2005)
Jackson, B., Jordán, T.: Globally rigid circuits of the direction–length rigidity matroid. J. Comb. Theory, Ser. B 100, 1–23 (2010)
Jackson, B., Jordán, T.: Brick partitions of graphs. Discrete Math. 310, 270–275 (2010)
Jackson, B., Jordán, T.: Operations preserving global rigidity of generic direction–length frameworks. Int. J. Comput. Geom. Appl. 20, 685–708 (2010)
Jackson, B., Keevash, P.: Necessary conditions for global rigidity of direction–length frameworks. Discrete Comput. Geom. (to appear)
Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)
Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discrete Math. 308, 1425–1437 (2008)
Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebr. Discrete Methods 3, 91–98 (1982)
Milnor, J.W.: Topology from the Differentiable Viewpoint. University Press of Virginia, Charlottesville (1965)
Nash-Williams, C.St.J.A.: Edge-disjoint spanning trees of finite graphs. J. Lond. Math. Soc. 36, 445–450 (1961)
Nash-Williams, C.St.J.A.: Decomposition of finite graphs into forests. J. Lond. Math. Soc. 39, 12 (1964)
Oxley, J.: Matroid Theory. Oxford University Press, London (1992)
Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proc. 17th Allerton Conference in Communications, Control and Computing, pp. 480–489 (1979)
Servatius, B., Whiteley, W.: Constraining plane configurations in CAD: Combinatorics of directions and lengths. SIAM J. Discrete Math. 12, 136–153 (1999)
Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 36, 221–230 (1961)
Whiteley, W.: Some matroids from discrete applied geometry. In: Matroid Theory. AMS Contemporary Mathematics, vol. 197, pp. 171–313 (1996)
Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM J. Discrete Math. 1, 237–255 (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by ERC grant 239696 and EPSRC grant EP/G056730/1.
Rights and permissions
About this article
Cite this article
Jackson, B., Keevash, P. Bounded Direction–Length Frameworks. Discrete Comput Geom 46, 48–71 (2011). https://doi.org/10.1007/s00454-011-9325-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-011-9325-0