This paper examines strong structural controllability of linear-time-invariant networked systems. We provide necessary and sufficient conditions for strong structural controllability involving constrained matchings over the bipartite graph representation of the network. An O(n²) algorithm to validate if a set of inputs leads to a strongly structurally controllable network and to find such an input set is proposed. The problem of finding such a set with minimal cardinality is shown to be NP-complete. Minimal cardinality results for strong and weak structural controllability are compared.