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Recently, graphical characterization of multiagent controllability has been studied extensively. A major effort in the study is to determine controllability directly from topology structures of communication graphs. In this paper, we proposed the concept of controllability destructive nodes, which indicates that the difficulty in graphical characterization turns out to be the identification of topology structures of controllability destructive nodes. It is shown that each kind of double and triple controllability destructive nodes happens to have a uniform topology structure which can be defined similarly. The definition, however, is verified not to be applicable to the topology structures of quadruple controllability destructive (QCD) nodes. Even so, a design method is proposed to uncover topology structures of QCD nodes for graphs with any size, and a complete graphical characterization is presented for the graphs consisting of five vertices. One advantage of the established complete graphical characterization is that the controllability of any graph with any selection of leaders can be determined directly from the identified/defined destructive topology structures. The results generate several necessary and sufficient graphical conditions for controllability. A key step of arriving at these results is the discovery of a relationship between the topology structure of the controllability destructive nodes and a corresponding eigenvector of the Laplacian matrix.
Ji et al. (Wed,) studied this question.