A scalable algorithm for the decomposition of minimally rigid graph

The rigid graph needs to be decomposed to solve the multi-equilibrium problem of the multi-agent formation control based on navigation function method. In this paper, a theorem and a scalable algorithm based on the Henneberg sequence of graphs are proposed for the decomposition of minimally rigid graph. The theorem demonstrates that if graph G( V, E) is a minimally rigid graph, then it can be decomposed as G = G t [Formula: see text] G c , where G t is a spanning tree of G and G c contains the remaining edges and their vertices. Moreover, the scalable algorithm is given to construct G t = ( V t , E t ) and G c = ( V c , E c ), and assign the edges in E c to n -- 2 distinct vertices in a scalable way via communication. Furthermore, a lemma is given to show when the number of vertices is less than eight, any edge can be chosen as the initialized edge, and the scalable algorithm mentioned above is always feasible. Finally, the effectiveness of the scalable algorithm is verified by numerical simulation.