We present a new polymial-time algorithm for finding minimal vertex covers in graphs. The algorithm finds a minimum vertex cover in all kwn examples of graphs. In view of the importance of the P versus NP question, we ask if there exists a graph for which the algorithm cant find a minimum vertex cover. The algorithm is demonstrated by finding minimum vertex covers for several famous graphs, including two large benchmark graphs with hidden minimum vertex covers. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows.