On Abstract Rigidity Matroids
Viet-Hang Nguyen
Source abstract
The problem of characterizing the generic rigidity matroid combinatorially is completely solved in dimension 2 but still open in higher dimensions. As a generalization of the generic rigidity matroid, Graver [SIAM J. Discrete Math., 4 (1991), pp. 355–368] introduced the concept of an abstract rigidity matroid. Answering an open question posed by Graver, Servatius, and Servatius [Combinatorial Rigidity, AMS, Providence, RI, 1993], this paper provides a combinatorial characterization of abstract rigidity matroids in any dimension. This combinatorial characterization leads to a polynomial algorithm for testing whether a given matroid is an abstract rigidity matroid. In dimension 2, the generic rigidity matroid can be viewed as an abstract rigidity matroid in which the independence is preserved under 1-extensions. This motivates us to introduce the concept of a 1-extendable abstract rigidity matroid, which is closer to the generic rigidity matroid. We show that in dimension 3, however, there exists a 1-extendable abstract rigidity matroid that is not a generic rigidity matroid.
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