Fundamental problems in control systems theory are controllability and observability, and designing control systems so that these properties are satisfied or approximated sufficiently. However, it is prudent to assume that an attacker will not only be able to subvert measurements but also control the system. Moreover, an advanced adversary with an understanding of the control system may seek to take over control of the entire system or parts thereof, or deny the legitimate operator this capability. The effectiveness of such attacks has been demonstrated in previous work. Indeed, these attacks cannot be ruled out given the likely existence of unknown vulnerabilities, increasing connectivity of nominally air-gapped systems and supply chain issues. The ability to rapidly recover control after an attack has been initiated and to detect an adversary’s presence is, therefore, critical. This paper focuses on the problem of structural controllability, which has recently attracted substantial attention through the equivalent problem of the power dominating set introduced in the context of electrical power network control. However, these problems are known to be
-hard with poor approximability. Given their relevance to many networks, especially power networks, this paper studies strategies for the efficient restoration of controllability following attacks and attacker-defender interactions in power-law networks.