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On the two-dimensional generic rigidity matroid and its dual

Brigitte Servatius
1991 Journal of combinatorial theory. Series B (Print)  
We give a characterization of the dual of the 2-dimensional generic rigidity matroid R(G) of a graph G and derive necessary and suffkient conditions for a connected matroid to be the rigidity matroid of  ...  It is well known, see [l] or [6] , that these independent edge sets are the independent sets of a matroid, the so-called 2-dimensional generic rigidity matroid, R(G), of the graph G.  ... 
doi:10.1016/0095-8956(91)90056-p fatcat:yoc5uyvfzjcqdh6646wrctg2ne

Page 1788 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews  
Raul Cordovil (P-INIC-AM) 92d:05042 05B35 05C10 52C25 Servatius, Brigitte (1-WPI) On the two-dimensional generic rigidity matroid and its dual. J. Combin. Theory Ser. B 53 (1991), no. 1, 106-113.  ...  The first major result in this paper is a characterization of the dual of the 2-dimensional generic rigidity matroid for (V, E).  ... 

Page 1788 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews  
Raul Cordovil (P-INIC-AM) 92d:05042 05B35 05C10 52C25 Servatius, Brigitte (1-WPI) On the two-dimensional generic rigidity matroid and its dual. J. Combin. Theory Ser. B 53 (1991), no. 1, 106-113.  ...  The first major result in this paper is a characterization of the dual of the 2-dimensional generic rigidity matroid for (V, E).  ... 

The universality theorem for neighborly polytopes [article]

Karim A. Adiprasito, Arnau Padrol
2014 arXiv   pre-print
In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of an even-dimensional neighborly polytope.  ...  This in particular provides the final step for Mn\"ev's proof of the universality theorem for simplicial polytopes.  ...  Also, we wish to thank Nikolai Mnëv, Jürgen Richter-Gebert and Bernd Sturmfels for helpful discussions concerning the history of the universality theorem, and the state of the universality theorem for  ... 
arXiv:1402.7207v2 fatcat:i6a5pemmqvh4lgfsrmfwtyalja

Dressians, tropical Grassmannians, and their rays

Sven Herrmann, Michael Joswig, David E Speyer
2014 Forum mathematicum  
These are also obtained by the two tropically rigid configuration in Figure 3 , where the left one gives 420 and the right one 560 rays.  ...  The tropical complex has two vertices, 0 and p , corresponding to the two maximal cells of the dual regular subdivision of ∆ k−1 × ∆ 1 .  ... 
doi:10.1515/forum-2012-0030 fatcat:xf5shxdognhyrcys3ddccepisq

Abstract 3-Rigidity and Bivariate C_2^1-Splines II: Combinatorial Characterization [article]

Katie Clinch, Bill Jackson, Shin-ichi Tanigawa
2021 arXiv   pre-print
We showed in the first paper of this series that the generic C_2^1-cofactor matroid is the unique maximal abstract 3-rigidity matroid.  ...  This solves the cofactor counterpart of the combinatorial characterization problem for the rigidity of generic 3-dimensional bar-joint frameworks.  ...  Acknowledgement We would like to thank Steve Noble for bringing our attention to the theory of matroid erections and for helpful conversations on this topic.  ... 
arXiv:1911.00207v2 fatcat:gckxkm5anjdwvep6pz6taisuma

Cryptomorphisms for abstract rigidity matroids [article]

Emanuele Delucchi, Tim Lindemann
2015 arXiv   pre-print
This note contributes to the structure theory of abstract rigidity matroids in general dimension.  ...  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  ...  Let U ⊆ V and let A be a m-dimensional abstract rigidity matroid on K(V ).  ... 
arXiv:1503.03795v1 fatcat:2m55i5nh6fbqlbexqzs5y6by3q

Generic Rigidity Matroids with Dilworth Truncations [article]

Shin-ichi Tanigawa
2012 arXiv   pre-print
We prove that the linear matroid that defines generic rigidity of d-dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained  ...  This leads to an alternative proof of Tay's combinatorial characterizations of generic rigidity of rod-bar frameworks and that of identified body-hinge frameworks.  ...  Acknowledgments The work was supported by Grant-in-Aid for JSPS Research Fellowships for Young Scientists.  ... 
arXiv:1010.5699v2 fatcat:vn5nv6oaxjevrl6ovmpycvenbi

Generic Rigidity Matroids with Dilworth Truncations

Shin-ichi Tanigawa
2012 SIAM Journal on Discrete Mathematics  
We cope with a more general structural model, body-rod-bar frameworks, and prove that the linear matroid defining its generic rigidity is equal to a counting matroid defined on the underlying graphs (Theorem  ...  They proved, as a new proof of Laman's theorem, that the linear matroid that defines the generic rigidity of two-dimensional bar-joint frameworks can be obtained from the union of two copies of a graphic  ...  The celebrated Laman's theorem [18] asserts that a two-dimensional bar-joint framework (Figure 1.1(a) ) is minimally rigid on a generic joint-configuration if and only if the graph G = (V, E) obtained  ... 
doi:10.1137/100819473 fatcat:55w5jkgi3baf7n3avnrepx5me4

On deformation and classification of V-systems [article]

V. Schreiber, A.P. Veselov
2014 arXiv   pre-print
We discuss also possible matroidal structures of V-systems in relation with projective geometry and give the catalogue of all known irreducible rank 3 V-systems.  ...  The V-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations.  ...  graphic matroids, their duals, and one special matroid on 10 elements.  ... 
arXiv:1404.4552v2 fatcat:pzq55nckh5e2niz3qjuscpsqni

Page 1889 of Mathematical Reviews Vol. , Issue 89D [page]

1989 Mathematical Reviews  
The two-frame, or union of two copies of the graphic matroid, is truncated to produce plane bar and joint frameworks giving a characterization of minimal infinitesimally rigid bar and joint frameworks  ...  Steiner system S(2,3,7) and S(3, 4,8), respectively; F} is the dual of F7; and Ag is obtained by deleting one edge from Kg.  ... 

Generate Δ-matroids from matroids [article]

Rémi Cocou Avohou, Brigitte Servatius, Herman Servatius
2021 arXiv   pre-print
We give necessary and sufficient conditions for two matroids on the same ground set to be the upper and lower matroid of a Δ-matroid.  ...  Let E = {a, b, c, d} and F be the subset of the power set of E containing M u = {{a, b, c, d}} and all the one element and two element subsets of E. Clearly D = (E, F) is a ∆-matroid.  ...  Adding to F all the three elements subsets of E gives the maximal ∆-matroid having the same lower matroid and upper matroid with D.  ... 
arXiv:2111.04035v1 fatcat:yki3qdccvjbrfbfmlzugchcv6i

On deformation and classification of ∨-systems

V. Schreiber, A.P. Veselov
2014 Journal of Nonlinear Mathematical Physics  
(called ∨-conditions) are satisfied for any α ∈ A and any two-dimensional plane Π ⊂ V * containing α and some ν, which may depend on Π and α.  ...  In the last section we study the property of the corresponding ν-function on the flats of matroid and state the uniqueness conjecture, saying that the matroid and function ν on its flats uniquely determine  ...  graphic matroids, their duals, and one special matroid on 10 elements.  ... 
doi:10.1080/14029251.2014.975528 fatcat:wvpxju3awrdztbyq5kogcajkz4

Neighborly Polytopes and Oriented Matroids

Bernd Sturmfels
1988 European journal of combinatorics (Print)  
Shemer: The chirotope of a neighborly 2k-polytope P is rigid, i.e. the entire internal structure of P is uniquely determined by its boundary complex.  ...  The theory of oriented matroids is applied to the class of neighborly convex polytopes.  ...  Let us see that its dual C{}j := X/ has the desired properties to be rigid but not neighborly: (l) ~ is clearly simplicial and realizable, since it is constructed from the realizable simplicial cyclic  ... 
doi:10.1016/s0195-6698(88)80050-7 fatcat:p32eo3u7obcz5dhotf5fxu3tcu

Page 1155 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
Together these two motivations lead to a general notion of triangulation of a vector arrangement, a notion which depends only on the oriented matroid of the arrangement.  ...  Aledo and José A. Galvez, Some rigidity results for compact spacelike surfaces in the 3-dimensional de Sitter space (19-27); Luis J. Alias and J.  ... 
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