Projective plane graphs and 3-rigidity

Abstract

It is shown that a simple graph which is embeddable in the real projective plane is minimally 3-rigid if and only if it is (3,6)-tight. Moreover the topologically uncontractible embedded graphs of this type are constructible from one of 8 embedded graphs by a sequence of vertex splitting moves. In particular the characterisation of minimal 3-rigidity holds for a triangulated Mobius strip.

34 pages (including a 9 page appendix). This is a substantial revision with fuller details and improved terminology. The main results are unchanged but there are new proofs and diagrams