Abstract
We prove the Molecular Conjecture posed by Tay and Whiteley. This implies that a graph G can be realized as an infinitesimally rigid panel-hinge framework in ℝd by mapping each vertex to a rigid panel and each edge to a hinge if and only if \(\bigl({d+1 \choose 2}-1\bigr)G\) contains \({d+1\choose2}\) edge-disjoint spanning trees, where \(\bigl({d+1 \choose2}-1\bigr)G\) is the graph obtained from G by replacing each edge by \(\bigl({d+1\choose2}-1\bigr)\) parallel edges.
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Amato, N.M.: The Protein Floding Server. Parasol, Texas A&M University, http://parasol.tamu.edu/groups/amatogroup/
Barnabei, M., Brini, A., Rota, G.: On the exterior calculus of invariant theory. J. Algebra 96(1), 120–160 (1985)
Catlin, P., Grossman, J., Hobbs, A., Lai, H.-J.: Fractional arboricity, strength, and principal partitions in graphs and matroids. Discrete Appl. Math. 40(3), 285–302 (1992)
Crapo, H., Whiteley, W.: Statics of frameworks and motions of panel structures: a projective geometric introduction. Struct. Topol. 6, 43–82 (1982)
Cunningham, W.: Optimal attack and reinforcement of a network. J. ACM 32(3), 549–561 (1985)
Doubilet, P., Rota, G.-C., Stein, J.: On the foundations of combinatorial theory IX. Combinatorial methods in invariant theory. Stud. Appl. Math. 53(3), 185–216 (1974)
Flexweb Server: http://flexweb.asu.edu
Graver, J.E., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics, vol. 2, p. 11. American Mathematical Society, New York (1993)
Gusfield, D.: Connectivity and edge-disjoint spanning trees. Inf. Process. Lett. 16(2), 87–89 (1983)
Jackson, B., Jordán, T.: Rank and independence in the rigidity matroid of molecular graphs. Technical Report TR-2006-02, Egerváry Research Group (2006)
Jackson, B., Jordán, T.: Rigid components in molecular graphs. Algorithmica 48(4), 399–412 (2007)
Jackson, B., Jordán, T.: Brick partitions of graphs. Discrete Math. 310(2), 270–275 (2008)
Jackson, B., Jordán, T.: On the rigidity of molecular graphs. Combinatorica 28(6), 645–658 (2008)
Jackson, B., Jordán, T.: Pin-collinear body-and-pin frameworks and the molecular conjecture. Discrete Comput. Geom. 40(2), 258–278 (2008)
Jackson, B., Jordán, T.: The generic rank of body–bar-and-hinge frameworks. Eur. J. Comb. 31(2), 574–588 (2009)
Jacobs, D.J., Kuhn, L.A., Thorpe, M.F.: Flexible and rigid regions in proteins. In: Rigidity Theory and Applications, pp. 357–384. Kluwer, Amsterdam (1999)
Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discrete Math. 308(8), 1425–1437 (2008)
Lei, M., Zavodszky, M., Kuhn, L., Thorpe, M.: Sampling protein conformations and pathways. J. Comput. Chem. 25(9), 1133–1148 (2004)
Lovász, L.: Combinatorial Problems and Exercises, 2nd edn. American Mathematical Society, New York (2007)
Nash-Williams, C.: Edge-disjoint spanning trees of finite graphs. J. Lond. Math. Soc. 1(1), 445–450 (1961)
Oxley, J.: Matroid Theory. Oxford University Press, Oxford (1992)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)
Selig, J.: Geometric Fundamentals of Robotics, 2nd edn. Springer, Berlin (2004)
Tay, T.: Linking (n−2)-dimensional panels in n-space II:(n−2,2)-frameworks and body and hinge structures. Graphs Comb. 5(1), 245–273 (1989)
Tay, T.S., Whiteley, W.: Recent advances in the generic rigidity of structures. Struct. Topol. 9, 31–38 (1984)
Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 36, 221–230 (1961)
Wells, S., Menor, S., Hespenheide, B., Thorpe, M.F.: Constrained geometric simulation of diffusive motion in proteins. Phys. Biol. 2, S127–S136 (2005)
White, N.: Grassmann-Cayley algebra and robotics. J. Intell. Robot. Syst. 11, 91–107 (1994)
White, N., Whiteley, W.: The algebraic geometry of motions of bar-and-body frameworks. SIAM J. Algebr. Discrete Methods 8(1), 1–32 (1987)
Whiteley, W.: Rigidity and polarity I: Statics of sheet structures. Geom. Dedic. 22, 329–362 (1987)
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. SIAM J. Discrete Math. 4, 75–95 (1989)
Whiteley, W.: Some matroids from discrete applied geometry. Contemp. Math. 197, 171–312 (1996)
Whiteley, W.: Rigidity of molecular structures: geometric and generic analysis. In: Thorpe, O., Duxbury, O. (eds.) Rigidity Theory and Applications, pp. 21–46. Kluwer, Amsterdam (1999)
Whiteley, W.: The equivalence of molecular rigidity models as geometric and generic graphs. Preprint (2004)
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. CRC Press, Boca Raton (2004), Chap. 60
Whiteley, W.: Counting out to the flexibility of molecules. Phys. Biol. 2, S116–S126 (2005)
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Katoh, N., Tanigawa, Si. A Proof of the Molecular Conjecture. Discrete Comput Geom 45, 647–700 (2011). https://doi.org/10.1007/s00454-011-9348-6
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DOI: https://doi.org/10.1007/s00454-011-9348-6