  This paper investigates the robustness of strong structural controllability
for linear time-invariant and linear time-varying directed networks with
respect to structural perturbations, including edge deletions and additions. In
this direction, we introduce a new construct referred to as a perfect graph
associated with a network with a given set of control nodes. The tight upper
bounds on the number of edges that can be added to, or removed from a network,
while ensuring strong structural controllability, are then derived. Moreover,
we obtain a characterization of critical edge-sets, the maximal sets of edges
whose any subset can be respectively added to, or removed from a network, while
preserving strong structural controllability. In addition, procedures for
combining networks to obtain strongly structurally controllable
network-of-networks are proposed. Finally, controllability conditions are
proposed for networks whose edge weights, as well as their structures, can vary
over time.
