668:
691:(two triangles sharing an edge) is rigid in two dimensions, but it is not uniquely realizable because it has two different realizations, one in which the triangles are on opposite sides of the shared edge and one in which they are both on the same side. Uniquely realizable graphs are important in applications that involve reconstruction of shapes from distances, such as
687:-dimensional space if every placement of the same graph with the same edge lengths is congruent to it. Such a framework must necessarily be rigid, because otherwise there exists a continuous motion bringing it to a non-congruent placement with the same edge lengths, but unique realizability is stronger than rigidity. For instance, the
359:, a vector for each vertex specifying its speed and direction. The gradient describes a linearized approximation to the actual motion of the points, in which each point moves at constant velocity in a straight line. The gradient may be described as a row vector that has one real number coordinate for each pair
658:
Although defined in different terms (column vectors versus row vectors, or forces versus motions) static rigidity and first-order rigidity reduce to the same properties of the underlying matrix and therefore coincide with each other. In two dimensions, the generic rigidity matroid also describes the
454:
If the edges of the framework are assumed to be rigid bars that can neither expand nor contract (but can freely rotate) then any motion respecting this rigidity must preserve the lengths of the edges: the derivative of length, as a function of the time over which the motion occurs, must remain zero.
1124:
showed that the same characterization is true: the independent sets form the edge sets of (2,3)-sparse graphs and the independent rigid sets form the edge sets of (2,3)-tight graphs. Based on this work the (2,3)-tight graphs (the graphs of minimally rigid generic frameworks in two dimensions) have
659:
number of degrees of freedom of a different kind of motion, in which each edge is constrained to stay parallel to its original position rather than being constrained to maintain the same length; however, the equivalence between rigidity and parallel motion breaks down in higher dimensions.
806:, redundantly rigid graph, and he conjectured that this is an exact characterization of the uniquely realizable frameworks. The conjecture is true for one and two dimensions; in the one-dimensional case, for instance, a graph is uniquely realizable if and only if it is connected and
1054:-sparse graph, for if not there would exist a subgraph whose number of edges would exceed the dimension of its space of equilibrium loads, from which it follows that it would have a self-stress. By similar reasoning, a set of edges that is both independent and rigid forms a
463:
of the rigidity matrix. For frameworks that are not in generic position, it is possible that some infinitesimally rigid motions (vectors in the nullspace of the rigidity matrix) are not the gradients of any continuous motion, but this cannot happen for generic frameworks.
1113:-tight graph. For instance, in one dimension, the independent sets form the edge sets of forests, (1,1)-sparse graphs, and the independent rigid sets form the edge sets of trees, (1,1)-tight graphs. In this case the rigidity matroid of a framework is the same as the
220:, an assignment of equal and opposite forces to the endpoints of each edge that is not identically zero but that adds to zero at every vertex. Thus, a set of edges forms an independent set in the rigidity matroid if and only if it has no self-stress.
575:, and when it does have this rank the only motions that preserve the lengths of the edges of the framework are the rigid motions. In this case the framework is said to be first-order (or infinitesimally) rigid. More generally, an edge
211:
is a special case of a load, in which equal and opposite forces are applied to the two endpoints of each edge (which may be imagined as a spring) and the forces formed in this way are added at each vertex. Every stress is an
216:, a load that does not impose any translational force on the whole system (the sum of its force vectors is zero) nor any rotational force. A linear dependence among the rows of the rigidity matrix may be represented as a
471:
to its original configuration. Rigid motions include translations and rotations of
Euclidean space; the gradients of rigid motions form a linear space having the translations and rotations as bases, of dimension
343:. If a set has this rank, it follows that its set of stresses is the same as the space of equilibrium loads. Alternatively and equivalently, in this case every equilibrium load on the framework may be
287:. An independent set in the rigidity matroid has a system of equilibrium loads whose dimension equals the cardinality of the set, so the maximum rank that any set in the matroid can have is
810:. However, Henrickson's conjecture is false for three or more dimensions. For frameworks that are not generic, it is NP-hard to determine whether a given framework is uniquely realizable.
1273:
1111:
1052:
764:
573:
341:
285:
1480:
Eren, T.; Goldenberg, O.K.; Whiteley, W.; Yang, Y.R.; Morse, A.S.; Anderson, B.D.O.; Belhumeur, P.N. (2004), "Rigidity, computation, and randomization in network localization",
519:, which must always be a subspace of the nullspace of the rigidity matrix. Because the nullspace always has at least this dimension, the rigidity matroid can have rank at most
517:
68:. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph.
1275:-tight graph is minimally rigid, and characterizing the minimally rigid graphs (the bases of the rigidity matroid of the complete graph) is an important open problem.
964:
932:
851:
800:
389:
993:
900:
171:
that has as its elements the edges of the graph. A set of edges is independent, in the matroid, if it corresponds to a set of rows of the rigidity matrix that is
1211:
1191:
1171:
1147:
871:
653:
633:
613:
593:
449:
429:
409:
1592:, DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 4, Providence, RI: American Mathematical Society, pp. 147–155,
459:
with the row of the rigidity matrix that represents the given edge. Thus, the family of gradients of (infinitesimally) rigid motions is given by the
766:
and that the matroid does not have any coloops. Hendrickson proved that every uniquely realizable framework (with generic edge lengths) is either a
995:
edges. From the consideration of loads and stresses it can be seen that a set of edges that is independent in the rigidity matroid forms a
455:
This condition may be expressed in linear algebra as a constraint that the gradient vector of the motion of the vertices must have zero
710:
if it remains rigid after removing any one of its edges. In matroidal terms, this means that the rigidity matroid has the full rank
700:
355:
If the vertices of a framework are in a motion, then that motion may be described over small scales of distance by its
1643:
1790:
1352:, Encyclopedia of Mathematics and its Applications, vol. 40, Cambridge: Cambridge Univ. Press, pp. 1–53,
347:
by a stress that generates an equal and opposite set of forces, and the framework is said to be statically rigid.
1548:
34:
1403:, Contemporary Mathematics, vol. 197, Providence, RI: American Mathematical Society, pp. 171–311,
451:-dimensional space; that is, the dimension of the gradient is the same as the width of the rigidity matrix.
1482:
Proc. Twenty-third Annual Joint
Conference of the IEEE Computer and Communications Societies (INFOCOM 2004)
1219:
1057:
998:
713:
522:
290:
234:
475:
1546:
Jackson, Bill; Jordán, Tibor (2005), "Connected rigidity matroids and unique realizations of graphs",
615:
if and only if there does not exist a continuous motion of the framework that changes the length of
1449:
803:
1795:
696:
1435:
Hendrickson, Bruce (1995), "The molecule problem: exploiting structure in global optimization",
1444:
467:
A rigid motion of the framework is a motion such that, at each point in time, the framework is
180:
807:
468:
187:
determine the same rigidity matroid, regardless of their specific coordinates. This is the (
1770:
1720:
1700:
1622:, Technical Report, Pittsburgh, PA: Computer Science Department, Carnegie-Mellon University
1597:
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93:
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8:
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22:
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113:
81:
38:
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1414:
1396:
1361:
1357:
1345:
1322:
1114:
207:
on a framework is a system of forces on the vertices (represented as vectors). A
42:
1489:
1173:
that forms a rigid framework in two dimensions, the spanning Laman subgraphs of
1638:
1562:
1409:
1150:
767:
168:
1762:
1666:
1784:
1611:
692:
688:
672:
456:
1688:
58:
1126:
695:
in land surveying, the determination of the positions of the nodes in a
152:
as endpoints, then the value of the entry is the difference between the
1712:
1511:
Hendrickson, Bruce (1992), "Conditions for unique graph realizations",
1641:; Theran, L. (2009), "Sparse hypergraphs and pebble game algorithms",
1657:
460:
1524:
1458:
1318:
1691:(1970), "On graphs and the rigidity of plane skeletal structures",
356:
1751:
International
Journal of Computational Geometry & Applications
231:, among which the equilibrium loads form a subspace of dimension
30:
667:
53:-dimensional space, a set of edges that defines a subgraph with
1149:
vertices forms the set of bases of the rigidity matroid of a
1744:"On the rank function of the 3-dimensional rigidity matroid"
699:, and the reconstruction of conformations of molecules via
183:
real numbers. Any two generic frameworks on the same graph
1479:
1399:(1996), "Some matroids from discrete applied geometry",
223:
The vector space of all possible loads, on a system of
1222:
1199:
1179:
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478:
437:
417:
397:
365:
293:
237:
431:
is the index of one of the
Cartesian coordinates of
96:
for each vertex of the graph. From a framework with
1267:
1205:
1185:
1165:
1141:
1105:
1046:
987:
958:
926:
894:
865:
845:
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758:
647:
627:
607:
595:belongs to the matroid closure operation of a set
587:
567:
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443:
423:
403:
383:
335:
279:
1256:
1235:
1094:
1073:
1035:
1014:
750:
729:
559:
538:
503:
482:
327:
306:
271:
250:
167:The rigidity matroid of the given framework is a
41:with rigid edges of fixed lengths, embedded into
1782:
1129:. The family of Laman graphs on a fixed set of
675:, generically rigid but not uniquely realizable
1741:
1637:
1625:
1545:
1305:Graver, Jack E. (1991), "Rigidity matroids",
818:
88:-dimensional Euclidean space by providing a
1510:
1434:
1430:
1428:
1391:
1389:
1387:
1385:
1383:
1381:
1379:
1377:
1375:
1656:
1590:Applied Geometry and Discrete Mathematics
1561:
1448:
1408:
1348:(1992), "Matroids and rigid structures",
1193:are the bases of the rigidity matroid of
45:. In a rigidity matroid for a graph with
1584:
1395:
1344:
1300:
1298:
1296:
1294:
1292:
1290:
1288:
1216:However, in higher dimensions not every
853:-sparse if every nonempty subgraph with
706:Bruce Hendrickson defined a graph to be
666:
1425:
1372:
813:
701:nuclear magnetic resonance spectroscopy
635:but leaves the lengths of the edges in
179:if the coordinates of its vertices are
18:Abstraction of bar-and-joint frameworks
1783:
1588:(1991), "On generic global rigidity",
1304:
1742:Jackson, Bill; Jordán, Tibor (2006),
1687:
1340:
1338:
1336:
1285:
1268:{\displaystyle (d,{\binom {d+1}{2}})}
1153:, and more generally for every graph
1121:
1106:{\displaystyle (d,{\binom {d+1}{2}})}
1047:{\displaystyle (d,{\binom {d+1}{2}})}
662:
1616:Embeddability of weighted graphs in
1610:
1307:SIAM Journal on Discrete Mathematics
759:{\displaystyle dn-{\binom {d+1}{2}}}
568:{\displaystyle dn-{\binom {d+1}{2}}}
336:{\displaystyle dn-{\binom {d+1}{2}}}
280:{\displaystyle dn-{\binom {d+1}{2}}}
112:columns, an expanded version of the
104:edges, one can define a matrix with
1484:, vol. 4, pp. 2673–2684,
120:. In this matrix, the entry in row
13:
1401:Matroid theory (Seattle, WA, 1995)
1333:
1239:
1077:
1018:
733:
542:
486:
310:
254:
191:-dimensional) rigidity matroid of
14:
1807:
1644:European Journal of Combinatorics
512:{\displaystyle {\binom {d+1}{2}}}
411:is a vertex of the framework and
1735:
1681:
1631:
1549:Journal of Combinatorial Theory
1604:
1578:
1539:
1504:
1473:
1262:
1223:
1100:
1061:
1041:
1002:
953:
941:
921:
909:
840:
828:
789:
777:
378:
366:
140:. If, on the other hand, edge
1:
1278:
350:
71:
33:that describes the number of
1437:SIAM Journal on Optimization
1358:10.1017/CBO9780511662041.002
1117:of the corresponding graph.
7:
1626:Jackson & Jordán (2005)
1490:10.1109/INFCOM.2004.1354686
819:Streinu & Theran (2009)
136:is not an endpoint of edge
10:
1812:
1693:J. Engineering Mathematics
1620:-space is strongly NP-hard
1563:10.1016/j.jctb.2004.11.002
198:
1763:10.1142/S0218195906002117
1667:10.1016/j.ejc.2008.12.018
1513:SIAM Journal on Computing
181:algebraically independent
966:-sparse and has exactly
821:define a graph as being
227:vertices, has dimension
175:. A framework is called
116:of the graph called the
1791:Mathematics of rigidity
697:wireless sensor network
57:degrees of freedom has
1410:10.1090/conm/197/02540
1269:
1207:
1187:
1167:
1143:
1107:
1048:
989:
960:
928:
896:
867:
847:
796:
760:
676:
649:
629:
609:
589:
569:
513:
445:
425:
405:
385:
337:
281:
21:In the mathematics of
1270:
1208:
1188:
1168:
1144:
1108:
1049:
990:
961:
959:{\displaystyle (k,l)}
929:
927:{\displaystyle (k,l)}
897:
873:vertices has at most
868:
848:
846:{\displaystyle (k,l)}
797:
795:{\displaystyle (d+1)}
761:
670:
650:
630:
610:
590:
570:
514:
446:
426:
406:
386:
384:{\displaystyle (v,i)}
338:
282:
94:Cartesian coordinates
1350:Matroid Applications
1220:
1197:
1177:
1157:
1133:
1125:come to be known as
1058:
999:
988:{\displaystyle kn-l}
970:
938:
906:
895:{\displaystyle kn-l}
877:
857:
825:
814:Relation to sparsity
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714:
639:
619:
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579:
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415:
395:
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235:
173:linearly independent
1705:1970JEnMa...4..331L
1120:In two dimensions,
64: −
23:structural rigidity
1713:10.1007/BF01534980
1265:
1203:
1183:
1163:
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681:unique realization
679:A framework has a
677:
663:Unique realization
645:
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605:
585:
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441:
421:
401:
381:
333:
277:
156:th coordinates of
35:degrees of freedom
1254:
1206:{\displaystyle G}
1186:{\displaystyle G}
1166:{\displaystyle G}
1142:{\displaystyle n}
1092:
1033:
866:{\displaystyle n}
748:
708:redundantly rigid
648:{\displaystyle S}
628:{\displaystyle e}
608:{\displaystyle S}
588:{\displaystyle e}
557:
501:
444:{\displaystyle d}
424:{\displaystyle i}
404:{\displaystyle v}
325:
269:
1803:
1775:
1773:
1757:(5–6): 415–429,
1748:
1739:
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1679:
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1651:(8): 1944–1964,
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1586:Connelly, Robert
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1397:Whiteley, Walter
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1346:Whiteley, Walter
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934:-tight if it is
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214:equilibrium load
114:incidence matrix
84:, embedded into
82:undirected graph
39:undirected graph
27:rigidity matroid
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118:rigidity matrix
74:
43:Euclidean space
19:
12:
11:
5:
1809:
1799:
1798:
1796:Matroid theory
1793:
1777:
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1734:
1699:(4): 331–340,
1680:
1630:
1624:. As cited by
1603:
1577:
1538:
1503:
1472:
1450:10.1.1.55.2335
1443:(4): 835–857,
1424:
1371:
1332:
1313:(3): 355–368,
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1151:complete graph
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768:complete graph
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169:linear matroid
73:
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9:
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4:
3:
2:
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1552:, Series B,
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1122:Laman (1970)
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1639:Streinu, I.
1612:Saxe, J. B.
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902:edges, and
655:unchanged.
218:self-stress
1785:Categories
1279:References
808:bridgeless
351:Kinematics
92:-tuple of
72:Definition
1729:122631794
1689:Laman, G.
1445:CiteSeerX
980:−
887:−
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469:congruent
461:nullspace
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199:Statics
177:generic
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