We address the two-machine flow-shop scheduling problem with time delays in order to minimize the makespan, i.e., the maximum completion time. We present a comprehensive theoretical analysis of the different lower bounds of the state of the art and we elucidate dominance relationships between them. We then introduce an exact method based on a branch-and-bound scheme. This method includes a local search-based heuristic and three dominance rules. Finally, a computer simulation of the branch-and-bound method is given on a set of 480 instances. We point out the good performance of our branch-and-bound method that outperforms the state of the art exact method. Precisely, we manage to solve all the sate of the art instances except one in a very short time.