  The Population Protocol model is a distributed model that concerns systems of
very weak computational entities that cannot control the way they interact. The
model of Network Constructors is a variant of Population Protocols capable of
(algorithmically) constructing abstract networks. Both models are characterized
by a fundamental inability to terminate. In this work, we investigate the
minimal strengthenings of the latter that could overcome this inability. Our
main conclusion is that initial connectivity of the communication topology
combined with the ability of the protocol to transform the communication
topology plus a few other local and realistic assumptions are sufficient to
guarantee not only termination but also the maximum computational power that
one can hope for in this family of models. The technique is to transform any
initial connected topology to a less symmetric and detectable topology without
ever breaking its connectivity during the transformation. The target topology
of all of our transformers is the spanning line and we call Terminating Line
Transformation the corresponding problem. We first study the case in which
there is a pre-elected unique leader and give a time-optimal protocol for
Terminating Line Transformation. We then prove that dropping the leader without
additional assumptions leads to a strong impossibility result. In an attempt to
overcome this, we equip the nodes with the ability to tell, during their
pairwise interactions, whether they have at least one neighbor in common.
Interestingly, it turns out that this local and realistic mechanism is
sufficient to make the problem solvable. In particular, we give a very
efficient protocol that solves Terminating Line Transformation when all nodes
are initially identical. The latter implies that the model computes with
termination any symmetric predicate computable by a Turing Machine of space
$\Theta(n^2)$.
