Combinatorial testing is an effective testing technique for detecting faults in a software or hardware system with multiple factors using combinatorial methods. By performing a test, which is an assignment of possible values to all the factors, and verifying whether the system functions as expected (pass) or not (fail), the presence of faults can be detected. The failures of the tests are possibly caused by combinations of multiple factors assigned with specific values, called faulty interactions. Martínez et al. [1] proposed the first deterministic adaptive algorithm for discovering faulty interactions involving at most two factors where each factor has two values, for which graph representations are adopted. In this paper, we improve Martínez et al.'s algorithm by an adaptive algorithmic approach for discovering faulty interactions in the so-called “non-2-locatable” graphs. We show that, for any system where each “non-2-locatable factor-component” involves two faulty interactions (for example, a system having at most two faulty interactions), our improved algorithm efficiently discovers all the faulty interactions with an extremely low mistaken probability caused by the random selection process in Martínez et al.'s algorithm. The effectiveness of our improved algorithm are revealed by both theoretical discussions and experimental evaluations.