Contingency analysis is an important building block in the stability and reliability analysis of power grid operations. However, due to the large number of transmission lines, in practice only a limited number of contingencies could be evaluated. This paper proposes a graph theory based N-1 contingency selection method to effectively identify the most critical contingencies, which can be used in security-constrained unit commitment (SCUC) problems to derive secure and economic operation decisions of the power grid. Specifically, the power flow transferring path identification algorithm and the vulnerability index are put forward to rank individual contingencies according to potential transmission line overloads. Effectiveness of the proposed N-1 contingency selection method is quantified by applying the corresponding SCUC solution to the full N-1 security evaluation, i.e., quantifying total post-contingency generation-load imbalance in all N-1 contingencies. Numerical results on several benchmark IEEE systems, including 5-bus, 24-bus, and 118-bus systems, show effectiveness of the proposed method. Compared with existing contingency selection methods which usually resort to full power flow calculations, the proposed method relies on power gird topology to effectively identify critical contingencies for facilitating the optimal scheduling of SCUC problems.