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This paper considers the structural controllability of a leader-follower multi-agent system. Graphical conditions for structural controllability based on the information flow graph of the system are provided. Then, the notions of p-link and q-agent controllability are introduced as quantitative measures for the controllability of the system subject to failure in communication links or agents. Necessary and sufficient conditions for the system to remain structurally controllable in the case of the failure of some of the communication links or loss of some agents are derived in terms of the topology of the information flow graph. Moreover, a polynomial-time algorithm for determining the maximum number of failed communication links under which the system remains structurally controllable is presented (which can be analogously developed for the case of agents loss). Finally, the proposed algorithm is extended to the case of loss of agents.