Isostatic bar and joint frameworks in the plane with irreducible pure conditions

Abstract

If a bar and joint framework in generic position is infinitesimally rigid with independent edges then we call it isostatic. This paper examines when the special positions that make a planar isostatic framework infinitesimally flexible form an irreducible variety. We find an inductive graph operation, called triangle-free edge splitting, which generates irreducible conditions. We introduce minimal isostatic graphs (MIG's), which are isostatic graphs that contain no proper isostatic subgraph. We settle the existence of an MIG for every number of vertices, except for v = 4, 5, 7.