On dimensional rigidity of bar-and-joint frameworks
Let V={1,2,,n} V = { 1 , 2 ,  , n } . A mapping p:V?Rr p : V ? R r , where p1,,pn p 1 ,  , p n are not contained in a proper hyper-plane is called an r r -configuration . Let G=(V,E) G = ( V , E ) be a simple connected graph on n n vertices. Then an r r -configuration p p together with graph G G , where adjacent vertices of G G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework ) in Rr R r , and is denoted by G(p) G ( p ) . In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) G ( p ) is dimensionally rigid. A given framework G(p) G ( p ) in Rr R r is said to be dimensionally rigid iff there does not exist a framework G(q) G ( q ) in Rs R s for s?r+1 s ? r + 1 , such that ?qi-qj?2=?pi-pj?2 ? q i - q j ? 2 = ? p i - p j ? 2 for all (i,j)?E ( i , j ) ? E . We present necessary and sufficient conditions for G(p) G ( p ) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p1,,pn p 1 ,  , p n of the given r r -configuration are in general position, is also investigated.
52c25 90c22 05c50 15a57