Abstract
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain “prescribed substructures”. We then prove a recursive version of this conjecture. (This is an extended version of the second author’s bachelor thesis at University of Bremen.).
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Notes
- 1.
In fact, there exists only one abstract 2-dimensional rigidity matroid on five vertices.
References
H.H. Crapo, G.-C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, preliminary edition (The M.I.T. Press, Cambridge, MA/London, 1970)
J.E. Graver, Rigidity matroids. SIAM J. Discrete Math. 4 (3), 355–368 (1991)
J. Graver, B. Servatius, H. Servatius, Combinatorial Rigidity. Graduate Studies in Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1993)
J.E. Graver, B. Servatius, H. Servatius, Abstract rigidity in m-space, in Jerusalem Combinatorics ’93. Contemporary Mathematics, vol. 178 (American Mathematical Society, Providence, RI, 1994), pp. 145–151
G. Laman, On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)
V.-H. Nguyen, On abstract rigidity matroids. SIAM J. Discrete Math. 24 (2), 363–369 (2010)
J. Oxley, Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. (Oxford University Press, Oxford, 2011)
S. Patkar, B. Servatius, K.V. Subrahmanyam, Abstract and generic rigidity in the plane. J. Combin. Theory Ser. B 62 (1), 107–113 (1994)
Acknowledgements
Emanuele Delucchi acknowledges partial support from SNSF-Professorship grant PP00P2_150552/1. Tim Lindemann acknowledges partial support from Swiss European Mobility Programme at University of Fribourg.
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Delucchi, E., Lindemann, T. (2016). Cryptomorphisms for Abstract Rigidity Matroids. In: Callegaro, F., Cohen, F., De Concini, C., Feichtner, E., Gaiffi, G., Salvetti, M. (eds) Configuration Spaces. Springer INdAM Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-31580-5_8
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DOI: https://doi.org/10.1007/978-3-319-31580-5_8
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