Zero forcing number, constraint matchings and strong structural controllability

The zero forcing number is a graph invariant introduced in order to study the minimum rank of the graph. In the first part of this paper, we first highlight that the computation of the zero forcing number of any directed graph (allowing loops) is NP-hard. Furthermore, we identify a class of directed trees for which the zero forcing number is computable in linear time. The second part of the paper is an application of the notion of zero forcing set in the study of the strong structural controllability of networked systems. This kind of controllability takes into account only the structure of the interconnection graph, but not the interaction strengths along the edges. We present the first efficient algorithm providing a minimum-size input set for the strong structural controllability of a self-damped or an undamped system with a tree structure.