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The zero forcing number is a graph invariant introduced to study the minimumrank of the graph. In 2008, Aazami proved the NP-hardness of computing the zeroforcing number of a simple undirected graph. We complete this NP-hardnessresult by showing that the non-equivalent problem of computing the zero forcingnumber of a directed graph allowing loops is also NP-hard. The rest of thepaper is devoted to the strong controllability of a networked system. This kindof controllability takes into account only the structure of the interconnectiongraph, but not the interconnection strengths along the edges. We provide anecessary and sufficient condition in terms of zero forcing sets for the strongcontrollability of a system whose underlying graph is a directed graph allowingloops. Moreover, we explain how our result differs from a recent related resultdiscovered by Monshizadeh et al. Finally, we show how to solve the problem offinding efficiently a minimum-size input set for the strong controllability ofa self-damped system with a tree-structure.

Zero forcing number, constrained matchings and strong structural controllability