Synthesis of Assur groups via group and matroid theory

The aim of this paper is to present a systematic approach for the structural synthesis of Assur groups from Baranov chains, based on group theory. An Assur group can be represented as a vertex-colored graph of a Baranov chain, in which a detached vertex of the graph is colored with a distinct color. To obtain the results presented herein, the synthesis of Baranov chains has been extended to planar rigid chains with up to 17 links. Thus, an algorithm based on group theory and graph theory is applied to obtain all Assur groups with up to 16 links. The detached vertices are selected by determining the orbits of the automorphism group of the graph, such that no isomorphic Assur groups are generated in the process. The results are compared with the values found in the literature, in order to verify possible inconsistencies and reconcile existing values.