Abstract
Given a Laman graph G, i.e. a minimally rigid graph in R 2, we provide a Θ(n 2) algorithm to augment G to a redundantly rigid graph, by adding a minimum number of edges. Moreover, we prove that this problem of augmenting is NP-hard for an arbitrary rigid graph G in R 2.
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Bang-Jensen, J., Gabow, H., Szigeti, Z., Jordán, T.: Edge-connectivity augmentation with partition constraints. SIAM J. Discrete Math. 12, 160–207 (1999)
Berg, A., Jordán, T.: A proof of Connelly’s conjecture on 3-connected circuits of the rigidity matroid. J. Comb. Theory, Ser. B 88, 77–97 (2003)
Berg, A., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Lecture Notes in Comput. Sci., vol. 2832, pp. 78–89. Springer, Berlin (2003)
Eren, T., Anderson, B., Morse, A., Whiteley, W., Belhumeur, P.: Operations on rigid formations of autonomous agents. Commun. Inf. Syst. 3, 223–258 (2004)
Eswaran, K.P., Tarjan, R.E.: Augmentation problems. SIAM J. Comput. 5, 653–665 (1976)
Fekete, Z.: Source location with rigidity and tree packing requirements. Oper. Res. Lett. 34, 607–612 (2006)
Fekete, Z., Jordán, T.: Rigid realizations of graphs on small grids. Comput. Geom. 32, 216–222 (2005)
Fekete, Z., Jordán, T.: Uniquely localizable networks with few anchors. In: Lecture Notes in Comput. Sci., vol. 4240, pp. 176–183. Springer, Berlin (2006)
Fekete, Z., Jordán, T., Whiteley, W.: An inductive construction for plane Laman graphs via vertex splitting. In: Lecture Notes in Comput. Sci., vol. 3221, pp. 299–310. Springer, Berlin (2004)
Frank, A. (ed.): Connectivity augmentation of networks: structures and algorithms. Math. Program. 84(3), 439–640 (1999)
Gabow, H., Jordán, T.: Incrementing bipartite digraph edge-connectivity. J. Comb. Opt. 4, 449–486 (2000)
Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1979)
Gluck, H.: Almost all simply connected closed surfaces are rigid. In: Lecture Notes in Math., vol. 438, pp. 225–239. Springer, Berlin (1975)
Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics, vol. 2. Am. Math. Soc., Providence (1993)
Haas, R., Orden, D., Rote, G., Santos, F., Servatius, B., Servatius, H., Souvaine, D., Streinu, I., Whiteley, W.: Planar minimally rigid graphs and pseudo-triangulations. Comput. Geom. 31(1–2), 31–61 (2005)
Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21(1), 65–84 (1992)
Henneberg, L.: Die graphische Statik der starren Systeme. Leipzig 1911, Johnson Reprint (1968)
Hsu, T.S.: On four-connecting a triconnected graph. In: Proc. 33rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 70–79 (1992)
Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory, Ser. B 94, 1–29 (2005)
Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. J. Comput. Phys. 137, 346–465 (1997)
Kant, G.: Augmenting outerplanar graphs. J. Algorithms 21, 1–25 (1996)
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 (k,l)-sparse graphs. In: EuroComb 2005, DMTCS Proc. AE 2005, pp. 181–186 (2005)
Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proceedings of the 17th Canadian Conference on Computational Geometry, CCCG’05, pp. 219–222 (2005)
Mourzakel, C.: An efficient algorithm for testing the generic rigidity of graphs in the plane. J. Phys. A 29(24), 8079–8098 (1996)
Olfati-Saber, R., Murray, R.M.: Graph rigidity and distributed formation stabilization of multi-vehicle systems. In: Proc. of the 41st Conference on Decision and Control, Las Vegas, NV, December 2002
Recski, A.: Matroid Theory and Its Applications. Springer, New York (1989)
Servatius, B.: Birigidity in the plane. SIAM J. Discrete Math. 2(4), 582–589 (1989)
Streinu, I.: Pseudo-triangulations, rigidity and motion planning. Discrete Comput. Geom. 34, 587–635 (2005)
Sugihara, K.: On some problems in the design of plane skeletal structures. SIAM J. Algorithms Discrete Methods 4(3), 355–362 (1983)
Thorpe, M.F., Duxbury, P.M. (eds.): Rigidity Theory and Applications. Kluwer Academic, Dordrecht (1999)
Watanabe, T., Nakamura, A.: Edge-connectivity augmentation problems. J. Comput. Syst. Sci. 35, 96–144 (1987)
Watanabe, T., Nakamura, A.: A smallest augmentation to 3-connect a graph. Discrete Appl. Math. 28, 183–186 (1990)
Whiteley, W.: Rigidity and scene analysis. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 1327–1354 (2004)
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Partially supported by Aragón Government under grant E58-DGA and MEC MTM2006-01267.
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García, A., Tejel, J. Augmenting the Rigidity of a Graph in R 2 . Algorithmica 59, 145–168 (2011). https://doi.org/10.1007/s00453-009-9300-9
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DOI: https://doi.org/10.1007/s00453-009-9300-9