Finding the Maximum Common Subgraph (MCS) in two graphs is NP-hard, and MCS algorithms adopt heuristics to improve time performance. In particular, the state-of-the-art branch-and-bound MCS algorithms, including McSplit, McSplit+RL, McSplit+LL, and McSplit+DAL, apply heuristics in selecting vertices to match from the two input graphs. However, they always select a vertex from the first graph to match with a set of candidate vertices from the second graph. This fixed order of the two graphs is not necessarily the best, because the number of candidates for a single vertex may vary. Therefore, we propose a bidirectional vertex selection method in which each time a single vertex is dynamically selected from either the first or the second graph depending on the number of candidate vertices to match in the other graph. This way the search space is reduced and the search time performance is improved. We have evaluated our method on a real-world chemical molecular graph dataset as well as other datasets. The results show that our proposed approach can find more non-trivial MCS than the best of McSplit family algorithms.