  This paper studies the controllability backbone problem in dynamical networks
defined over graphs. The main idea of the controllability backbone is to
identify a small subset of edges in a given network such that any subnetwork
containing those edges/links has at least the same network controllability as
the original network while assuming the same set of input/leader vertices. We
consider the strong structural controllability (SSC) in our work, which is
useful but computationally challenging. Thus, we utilize two lower bounds on
the network's SSC based on the zero forcing notion and graph distances. We
provide algorithms to compute controllability backbones while preserving these
lower bounds. We thoroughly analyze the proposed algorithms and compute the
number of edges in the controllability backbones. Finally, we compare and
numerically evaluate our methods on random graphs.
