Ensuring reliable operation of large power systems subjected to multiple outages is a challenging task because of the combinatorial nature of the problem. Traditional methods of steady-state security assessment in power systems involve contingency analysis based on AC or DC power flows. However, power flow based contingency analysis is not fast enough to evaluate all contingencies for real-time operations. Therefore, real-time contingency analysis (RTCA) only evaluates a subset of the contingencies (called the contingency list), and hence might miss critical contingencies that lead to cascading failures.This dissertation proposes a new graph-theoretic approach, called the feasibility test (FT) algorithm, for analyzing whether a contingency will create a saturated or over-loaded cut-set in a meshed power network; a cut-set denotes a set of lines which if tripped separates the network into two disjoint islands. A novel feature of the proposed approach is that it lowers the solution time significantly making the approach viable for an exhaustive real-time evaluation of the system. Detecting saturated cut-sets in the power system is important because they represent the vulnerable bottlenecks in the network. The robustness of the FT algorithm is demonstrated on a 17,000+ bus model of the Western Interconnection (WI). Following the detection of post-contingency cut-set saturation, a two-component methodology is proposed to enhance the reliability of large power systems during a series of outages. The first component combines the proposed FT algorithm with RTCA to create an integrated corrective action (iCA), whose goal is to secure the power system against post-contingency cut-set saturation as well as critical branch overloads. The second component only employs the results of the FT to create a relaxed corrective action (rCA) that quickly secures the system against saturated cut-sets. The first component is more comprehensive than the second, but the latter is computationally more efficient. The effectiveness of the two components is evaluated based upon the number of cascade triggering contingencies alleviated, and the computation time. Analysis of different case-studies on the IEEE 118-bus and 2000-bus synthetic Texas systems indicate that the proposed two-component methodology enhances the scope and speed of power system security assessment during multiple outages.