This paper addresses a problem of path planning for multiple robots. An abstraction where the environment for robots is modeled as an undirected graph with robots placed in its vertices is used (this abstraction is also known as the problem of pebble motion on graphs). A class of the problem with bi-connected graph and at least two unoccupied vertices is defined. A novel polynomial-time solution algorithm for this class of problem is proposed. It is shown in the paper that the new algorithm significantly outperforms the existing state-of-the-art techniques applicable to the problem. Moreover, the performed experimental evaluation indicates that the new algorithm scales up well which make it suitable for practical problem solving.