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This paper presents an innovative approach to maximally disconnect a given network. More specifically, this work introduces the concept of a Critical Disruption Path, a path between a source and a destination vertex whose deletion minimizes the cardinality of the largest remaining connected component. Network interdiction models seek to optimally disrupt network operations. Existing interdiction models disrupt network operations by removing vertices or edges. We introduce the first problem and formulation that optimally fragments a network via interdicting a path. Areas of study in which this work can be applied include transportation and evacuation networks, surveillance and reconnaissance operations, anti-terrorism activities, drug interdiction, and counter human-trafficking operations. In this paper, we first address the complexity associated with the Critical Disruption Path problem, and then provide a Mixed-Integer Linear Programming formulation for finding its optimal solution. Further, we develop a tailored Branch-and-Price algorithm that efficiently solves the Critical Disruption Path problem. We demonstrate the superiority of the developed Branch-and-Price algorithm by comparing the results found via our algorithm with the results found via the monolith formulation. In more than half of the test instances that can be solved by both the monolith and our Branch-and-Price algorithm, we outperform the monolith by two orders of magnitude.