  The topology of a robotic swarm affects the convergence speed of consensus
and the mobility of the robots. In this paper, we prove the existence of a
complete set of local topology manipulation operations that allow the
transformation of a swarm topology. The set is complete in the sense that any
other possible set of manipulation operations can be performed by a sequence of
operations from our set. The operations are local as they depend only on the
first and second hop neighbors' information to transform any initial spanning
tree of the network's graph to any other connected tree with the same number of
nodes. The flexibility provided by our method is similar to global methods that
require full knowledge of the swarm network. We prove the existence of a
sequence of transformations for any tree-to-tree transformation, and derive
sequences of operations to form a line or star from any initial spanning tree.
Our work provides a theoretical and practical framework for topological control
of a swarm, establishing global properties using only local information.
