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In this paper, several necessary and sufficient graphical conditions arederived for the controllability of multi-agent systems by taking advantage ofthe proposed concept of controllability destructive nodes. A key step ofarriving at this result is the establishment of a relationship between topologystructures of the controllability destructive nodes and a specific eigenvectorof the Laplacian matrix. The results on double, triple and quadruplecontrollability destructive nodes constitute a novel approach to study thecontrollability. In particular, the approach is applied to the graph consistingof five nodes to get a complete graphical characterization of controllability.

Destructive nodes in multi-agent controllability