A two-parameter continuation algorithm is developed for combustion processes which can calculate exact (within some tolerance) ignition and extinction conditions and provide sensitivity of ignition and extinction solutions on chemical and transport/fluid mechanics parameters. In addition, two-parameter bifurcation diagrams of ignition and extinction conditions versus any parameter (e.g., pressure) can be computer-generated. The new algorithm surmounts limitations of previous simulations, where approximate critical conditions for flame stability were determined, and extends sensitivity of regular solutions to critical solutions having a singular Jacobian. In conjunction with principal component analysis, this algorithm provides a systematic methodology for mechanism reduction at ignition and extinction. As an example, this methodology is applied to ignition and extinction of hydrogen/air mixtures in a jet-stirred reactor. Important reactions at ignition, extinction, and for flame multiplicity are identified. Two minimal schemes of six reactions are found to lead to flame multiplicity. The validity of quasi-steady-state approximation for all species at ignition and extinction is also examined as a function of pressure. For example, it is shown that on the second branch of the pressure-temperature (P-T) ignition diagram, H, O, and OH are in quasi-steady-state. Predictions from proposed ignition and extinction criteria are in good agreement with detailed kinetics simulations.