  In this paper, we study the construction and transformation of
two-dimensional persistent graphs. 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 distances constraints seeking to preserve a
formation shape. Analogously to the powerful results about Henneberg sequences
in minimal rigidity theory, we propose different types of directed graph
operations allowing one to sequentially build any minimally persistent graph
(i.e. persistent graph with a minimal number of edges for a given number of
vertices), each intermediate graph being also minimally persistent. We also
consider the more generic problem of obtaining one minimally persistent graph
from another, which corresponds to the on-line reorganization of an autonomous
agent formation. We prove that we can obtain any minimally persistent formation
from any other one by a sequence of elementary local operations such that
minimal persistence is preserved throughout the reorganization process.
