Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Symmetric Laman theorems for the groups C2 and Cs. | Symmetric Laman theorems for the groups C2 and Cs Bernd Schulze Institute of Mathematics MA 6-2 TU Berlin 10623 Berlin Germany bschulze@ Submitted Jun 17 2010 Accepted Nov 3 2010 Published Nov 19 2010 Mathematics Subject Classifications 52C25 70B99 05C99 Abstract For a bar and joint framework G p with point group C3 which describes 3-fold rotational symmetry in the plane it was recently shown in Schulze Discret. Comp. Geom. 44 946-972 that the standard Laman conditions together with the condition derived in Connelly et al. Int. J. Solids Struct. 46 762-773 that no vertices are fixed by the automorphism corresponding to the 3-fold rotation geometrically no vertices are placed on the center of rotation are both necessary and sufficient for G p to be isostatic provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In this paper we prove the analogous Laman-type conjectures for the groups C2 and Cs which are generated by a half-turn and a reflection in the plane respectively. In addition analogously to the results in Schulze Discret. Comp. Geom. 44 946-972 we also characterize symmetry generic isostatic graphs for the groups C2 and Cs in terms of inductive Henneberg-type constructions as well as Crapo-type 3Tree2 partitions - the full sweep of methods used for the simpler problem without symmetry. 1 Introduction The major problem in generic rigidity is to find a combinatorial characterization of graphs whose generic realizations as bar-and-joint frameworks in d-space are rigid. While for dimension d 3 only partial results for this problem have been found it is completely solved for dimension 2. In fact using a number of both algebraic and combinatorial techniques a series of characterizations of generically rigid graphs in the plane have been established ranging from Laman s famous counts from 1970 on the number of vertices Research for this article was supported in part under a grant from NSERC .
