Minimum leader selection for controllability of undirected graphs with leader-follower framework

Researchers are paying increased attention to the controllability of undirected networks owing to their broad scope of application. This paper addresses the optimization problem of the minimum set of leaders for the controllability of undirected graphs. Existing results show that this is a challenging problem. It is difficult to find not only its optimal solution but also its approximate algorithm. A new concept is proposed, namely minimal perfect critical set (MPCS), to obtain an optimal solution with high probability from a purely graph-theoretic perspective. Some properties are presented and based on these theorems, the problem of the minimum set of leaders of two typical self-similar bipartite networks, namely deterministic scale-free networks (DSFN) and Cayley trees, is solved completely.