Rigid Graph
The word "rigid" has two different meaning when applied to a graph. Firstly, a rigid graph may refer to a graph having a graph
automorphism group containing a single element.
A framework (or graph) is rigid iff continuous motion of the points of the configuration
maintaining the bar constraints comes from a family of motions of all Euclidean
space which are distance-preserving. A graph that is
not rigid is said to be flexible (Maehara 1992).
For example, the cycle graph 
is rigid, while
is flexible. An embedding of the bipartite
graph 
in the plane is rigid unless its
six vertices lie on a conic (Bolker and Roth 1980,
Maehara 1992).
A graph 
is (generically)
-rigid if, for almost all (i.e., an open dense set
of) configurations of 
, the framework 
is rigid in 
.
Cauchy (1813) proved the rigidity theorem, one of the first results in rigidity theory. Although rigidity problems were of immense
interest to engineers, intensive mathematical study of these types of problems has
occurred only relatively recently (Connelly 1993, Graver et al. 1993).
SEE ALSO: Braced Polygon,
Flexible Graph,
Flexible Polyhedron,
Framework,
Graph Bar,
Harborth Graph,
Just Rigid,
Laman's Theorem,
Liebmann's Theorem,
Rigid
Polyhedron,
Rigidity Theorem,
Tensegrity
REFERENCES:
Asimov, L. and Roth, B. "The Rigidity of Graphs." Trans. Amer. Math.
Soc. 245, 279-289, 1978.
Bolker, E. D. and Roth, B. "When is a Bipartite Graph a Rigid Framework?"
Pacific J. Math. 90, 27-44, 1980.
Cauchy, A. L. "Sur les polygones et les polyèdres." XVIe
Cahier IX, 87-89, 1813.
Connelly, R. "Rigidity." Ch. 1.7 in Handbook of Convex Geometry, Vol. A (Ed. P. M. Gruber and J. M. Wills).
Amsterdam, Netherlands: North-Holland, pp. 223-271, 1993.
Coxeter, H. S. M. and Greitzer, S. L. Geometry
Revisited. Washington, DC: Math. Assoc. Amer., p. 56, 1967.
Crapo, H. and Whiteley, W. "Statics of Frameworks and Motions of Panel Structures, A Projective Geometry Introduction." Structural Topology 6, 43-82,
1982.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Rigidity of Frameworks." §B14 in Unsolved
Problems in Geometry. New York: Springer-Verlag, pp. 63-65, 1991.
Dehn, M. "Über die Strakheit knovexer Polyeder." Math. Ann. 77,
466-473, 1916.
Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51,
165-170, 1978.
Graver, J.; Servatius, B.; and Servatius, H. Combinatorial
Rigidity. Providence, RI: Amer. Math. Soc., 1993.
Maehara, H. "Distance Graphs in Euclidean Space." Ryukyu Math. J. 5,
33-51, 1992.
Pegg, E. Jr. "Rigid Nonagon." http://www.mathpuzzle.com/riginona.gif.
Roth, B. "Rigid and Flexible Frameworks." Amer. Math. Monthly 88,
6-21, 1981.
Referenced on Wolfram|Alpha:
Rigid Graph
CITE THIS AS:
Weisstein, Eric W. "Rigid Graph." From
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