  This paper treats the problem of the merging of formations, where the
underlying model of a formation is graphical. We first analyze the rigidity and
persistence of meta-formations, which are formations obtained by connecting
several rigid or persistent formations. Persistence is a generalization to
directed graphs of the undirected notion of rigidity. In the context of moving
autonomous agent formations, persistence characterizes the efficacy of a
directed structure of unilateral distance constraints seeking to preserve a
formation shape. We derive then, for agents evolving in a two- or
three-dimensional space, the conditions under which a set of persistent
formations can be merged into a persistent meta-formation, and give the minimal
number of interconnections needed for such a merging. We also give conditions
for a meta-formation obtained by merging several persistent formations to be
persistent.
