Abstract
We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley’s direction networks.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brylawski, T.: Constructions. In: White, N. (ed.) Theory of Matroids, Encyclopedia of Mathematics and Its Applications, pp. 127–223. Cambridge University Press, Cambridge (1986)
Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (1997)
Fekete, Z.: Source location with rigidity and tree packing requirements. Oper. Res. Lett. 34(6), 607–612 (2006)
Gluck, H.: Almost all simply connected closed surfaces are rigid. Lect. Notes Math. 438, 225–239 (1975)
Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompositions. J. Comb. Math. Comb. Comput. 62, 3–11 (2007)
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(8), 1425–1437 (2008). doi:10.1016/j.disc.2007.07.104
Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007)
Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007)
Lovász, L.: Combinatorial Problems and Exercises, 2nd edn. AMS Chelsea, Providence (2007)
Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebr. Discrete Methods 3(1), 91–98 (1982)
Lovász, L.: Matroid matching and some applications. J. Comb. Theory, Ser. B 28, 208–236 (1980)
Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27, 294 (1864)
Oxley, J.G.: Matroid Theory. Clarendon/Oxford University Press, New York (1992)
Recski, A.: Matroid Theory and Its Applications in Electric Network Theory and in Statics. Springer, Berlin (1989)
Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly np-hard. In: Proc. of 17th Allerton Conference in Communications, Control, and Computing, Monticello, IL, pp. 480–489 (1979)
Streinu, I., Theran, L.: Sparsity-certifying graph decompositions. Graphs Comb. 25(2), 219–238 (2009). doi:10.1007/s00373-008-0834-4
Tay, T.S.: A new proof of Laman’s theorem. Graphs Comb. 9, 365–370 (1993)
Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM J. Discrete Math. 1(2), 237–255 (1988)
Whiteley, W.: A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom. 4, 75–95 (1989)
Whiteley, W.: Some matroids from discrete applied geometry. In: Bonin, J., Oxley, J.G., Servatius, B. (eds.) Matroid Theory. Contemporary Mathematics, vol. 197, pp. 171–311. American Mathematical Society, Providence (1996)
Whiteley, W.: Rigidity and scene analysis. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 1327–1354. CRC Press, Boca Raton (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Streinu, I., Theran, L. Slider-Pinning Rigidity: a Maxwell–Laman-Type Theorem. Discrete Comput Geom 44, 812–837 (2010). https://doi.org/10.1007/s00454-010-9283-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-010-9283-y