Rigid Two-Dimensional Frameworks with Three Collinear Points
Tóm tắt
Let G = (V, E) be a graph and x, y, z ∈ V be three designated vertices. We give a necessary and sufficient condition for the existence of a rigid two-dimensional framework (G, p), in which x, y, z are collinear. This result extends a classical result of Laman on the existence of a rigid framework on G. Our proof leads to an efficient algorithm which can test whether G satisfies the condition.
Tài liệu tham khảo
Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Di Battista, G., Zwick, U. (eds) Proceedings 11th annual european symposium on algorithms (ESA) 2003, Springer lecture notes in computer science 2832, 2003, pp. 78–89
Bolker, E.D., Roth, B.: When is a bipartite graph a rigid framework? Pacific J. Math. 90, 27–44 (1980)
Crapo, H., Whiteley, W.: Statics of frameworks and motions of panel structures, a projective geometric introduction. Shape Structural Topology 6, 43–82 (1982)
Gluck, H.: Almost all simply connected closed surfaces are rigid. In: Geometric Topology Proceedings of the Conference, Park City, Utah, 1974, Lecture Notes in Mathematics, vol. 438, Springer, Berlin, 1975, pp. 225–239
Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity, AMS Graduate Studies in Mathematics Vol. 2, 1993
Henneberg, L.: Die graphische Statik der starren Systeme, Leipzig, 1911
Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Combin. Theory Ser. B. 94, 1–29 (2005)
Laman, G.: On graphs and rigidity of plane skeletal structures. J. Engineering Math. 4, 331–340 (1970)
Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebraic Discrete Methods 3(1), 91–98 (1982)
Oxley, J.G.: Matroid theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992, xii+532
Tay, T.S., Whiteley, W.: Generating isostatic frameworks. Structural Topology 11, 21–69 (1985)
White, N.L., Whiteley, W.: The algebraic geometry of stresses in frameworks. SIAM J. Alg. Disc. Meth. 4, 481–511 (1983)
Whiteley, W.: Infinitesimal motions of bipartite frameworks. Pacific J. Math. 110, 233–255 (1984)
Whiteley, W.: Some matroids from discrete applied geometry. Matroid theory (Seattle, WA, 1995), pp. 171–311; Contemp. Math., 197, Am. Math. Soc., Providence, RI, 1996