Gain-Sparsity and Symmetry-Forced Rigidity in the Plane
We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral sym...
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| Published in | Discrete & computational geometry Vol. 55; no. 2; pp. 314 - 372 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Springer US
01.03.2016
Springer Nature B.V |
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| Abstract | We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2
k
with odd
k
, unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph. |
|---|---|
| AbstractList | We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2
k
with odd
k
, unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph. We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2 with odd , unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph. We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph. |
| Author | Tanigawa, Shin-ichi Kaszanitzky, Viktória E. Jordán, Tibor |
| Author_xml | – sequence: 1 givenname: Tibor surname: Jordán fullname: Jordán, Tibor organization: Department of Operations Research, Eötvös University, and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization – sequence: 2 givenname: Viktória E. surname: Kaszanitzky fullname: Kaszanitzky, Viktória E. organization: Department of Operations Research, Eötvös University, Department of Mathematics and Statistics, Lancaster University – sequence: 3 givenname: Shin-ichi surname: Tanigawa fullname: Tanigawa, Shin-ichi email: tanigawa@kurims.kyoto-u.ac.jp organization: Research Institute for Mathematical Sciences, Kyoto University, Centrum Wiskunde & Informatica (CWI) |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/26900195$$D View this record in MEDLINE/PubMed |
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| CitedBy_id | crossref_primary_10_1016_j_ifacol_2023_10_301 crossref_primary_10_1016_j_jmaa_2020_124895 crossref_primary_10_1137_20M1346961 crossref_primary_10_1016_j_ejc_2022_103639 crossref_primary_10_1016_j_jsc_2018_02_004 crossref_primary_10_1007_s00454_019_00135_5 crossref_primary_10_1016_j_ijsolstr_2023_112492 crossref_primary_10_1007_s00493_016_3469_8 crossref_primary_10_1007_s00454_020_00198_9 crossref_primary_10_1016_j_jctb_2020_09_009 crossref_primary_10_1016_j_laa_2020_08_004 crossref_primary_10_1137_17M1116088 |
| Cites_doi | 10.1088/0953-8984/19/40/406218 10.1016/0095-8956(91)90005-5 10.1007/978-3-540-39658-1_10 10.1016/0020-7683(80)90087-6 10.1016/j.jctb.2004.11.002 10.1007/s00454-015-9692-z 10.1088/1478-3975/2/4/S06 10.1016/j.disc.2007.07.104 10.1016/j.aim.2012.10.007 10.1137/130947192 10.1137/090776238 10.1007/s10711-013-9878-6 10.1142/S0218195910003505 10.1007/s00454-010-9317-5 10.1098/rspa.2009.0676 10.1007/BF01177190 10.3390/sym6030516 10.1007/s00454-009-9231-x 10.1023/A:1015366416311 10.1016/0095-8956(89)90063-4 10.1090/tran/6401 |
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| Keywords | Frame matroids Infinitesimal rigidity Frameworks Primary 52C25 Group-labeled graphs Secondary 05B35 68R10 Rigidity matroids Symmetry Rigidity of graphs |
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| References_xml | – reference: KannoYOhsakiMMurotaKKatohNGroup symmetry in interior-point methods for semidefinite programOpt. Eng.2001229332010.1023/A:101536641631119802761035.90056 – reference: SchulzeBTanigawaSInfinitesimal rigidity of symmetric frameworksSIAM J. Discrete Math.20152931259128610.1137/1309471923374639 – reference: SchulzeBWhiteleyWThe orbit rigidity matrix of a symmetric frameworkDiscrete Comput. Geom.201146356159810.1007/s00454-010-9317-528269701235.05040 – reference: TanigawaSMatroids of gain graphs in discrete applied geometryTrans. Am. Math. Soc.201536720158597864110.1090/tran/64013403067 – reference: TarnaiTSimultaneous static and kinematic indeterminacy of space trusses with cyclic symmetryInt. J. Solids Struct.19801634735910.1016/0020-7683(80)90087-60433.73043 – reference: WhiteleyWCounting out to the flexibility of moleculesPhys. 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| SubjectTerms | Combinatorial analysis Combinatorics Computational geometry Computational Mathematics and Numerical Analysis Gain Geometry Graph theory Graphs Mathematics Mathematics and Statistics Planes Rigidity Symmetry |
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| Title | Gain-Sparsity and Symmetry-Forced Rigidity in the Plane |
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