Sorting Separable Permutations by Restricted Multi-break Rearrangements

A multi-break rearrangement generalizes most of genome rearrangements, such as block-interchanges, transpositions and reversals. A k-break cuts k adjacencies over a permutation, and forms k new adjacencies by joining the extremities according to an arbitrary matching. Block-interchange distance is a polynomial problem, but the transposition and the reversal distances are both NP-hard problems. A FPT algorithm is known for the multi-break distance between two permutations. We propose the restricted multi-break rearrangement (rmb), where a restricted k-break cuts k adjacencies but forms k new adjacencies according to a fixed matching. By considering permutations graphs we are able to formulate a better way to represent the orders of a permutations. Cographs are P4-free graphs, a subclass of permutation graphs. The permutations that characterize cographs are the separable permutations, exactly the permutations which do not contain particular patterns that yield P4’s. By using their cotree representation, we give an algorithm to sort by rmb the separable permutations. 2000 AMS Subject Classification: 68Q17, 68P10 and 05A05.