Leírás
A d-dimensional framework is a pair (G, p), where G=(V, E) is a graph and p is a map from V to the d-dimensional Euclidean space. An infinitesimal motion of (G, p) is another map from V to R^d such that moving each point of the framework in that direction does not change the distances corresponding to edges in the first order. The framework is infinitesimally rigid if all of its infinitesimal motions correspond to some isometries of R^d.
Laman (1970) characterized the infinitesimal rigidity of bar-joint frameworks in the plane when the framework is in generic position, that is, when the coordinates of the points are algebraically independent over the field of rationals. Adiprasito and Nevo (2018) recently asked the following question: Which graph classes have infinitesimally rigid realizations for each of its members on a fixed constant number of points in R^d. They showed that triangulated planar graphs have such realizations on 76 points in R^3, however, for each constant c and for d>1, there always exists a graph in the class of generically rigid graphs in R^d that cannot be realized as an infinitesimally rigid bar-joint framework on any c points in R^d.
Based on the above results, it is a natural question whether planar graphs which are generically rigid in the plane have an infinitesimally rigid realization on a constant number of points of the plane. The main result of my talk is that every planar graph which is generically rigid in the plane has an infinitesimally rigid realization on 26 points of the plane. Moreover, given any set of 26 points in the plane such that the coordinates of the points are algebraically independent over the field of rationals, one can find an infinitesimally rigid realization of any rigid planar graph on that set.