This paper presents a contingency screening method and a framework for its on-line implementation. The proposed method carries out contingency screening and on-line stability assessment with respect to first-swing transient stability. For that purpose, it utilizes the single machine equivalent method and aims at improving the prior developed contingency screening approaches. In order to determine vulnerability of the system with respect to a particular contingency, only one time-domain simulation needs to be performed. An early stop criteria is proposed so that in a majority of the cases the simulation can be terminated after a few hundred milliseconds of simulated system response. The method's outcome is an assessment of the system's stability and a classification of each considered contingency. The contingencies are categorized by exploiting parameters of an equivalent one machine infinite bus system. A novel island detection approach, appropriate for an on-line application since it utilizes efficient algorithms from graph theory and enables stability assessment of individual islands, is also introduced. The New England and New York system as well as the large-scale model of the Continental-European interconnected system are used to test the proposed method with respect to assessment accuracy and computation time.