Super oriented cycles in permutations

Abstract

The degree of dissimilarity between genome sequences of homologous species is a measure of the evolutionary distance between them. It serves as a metric in the construction of phylogenetic trees, which depict the evolutionary relationships and common ancestry among different species. Given two genome sequences, evolutionary distance is determined by estimating the number of global mutations that transform one sequence to the other. The computation of the evolutionary distance is done by modelling a genome with the corresponding permutation. Global rearrangement operations such as transposition that model a particular genomic mutation are studied by employing a combinatorial structure known as a cycle graph of the corresponding permutation. A cycle in a cycle graph that has odd length is called an odd cycle. In the context of the problem of sorting by transpositions (SBT), a valid 2-move is a transposition that increases the number of odd cycles in the cycle graph by two. A super oriented cycle (SOC) is an odd cycle C where C and one of the resultant cycles admit valid 2-moves. The minimum number of mutations required to transform a species S into a related species T is the distance from S to T under that mutation. Christie opined that characterizing SOCs will improve the lower bound of the transposition distance. We characterize super oriented cycles. Equivalent transformations on permutations like reduction and (g,b)-split preserve the transposition distance of a given permutation and map SBT to the corresponding SBT on a transformed simpler permutation. We introduce merge, a novel equivalent transformation. These results have applications in computing transposition and other distances between related species.

Keywords: Approximation algorithms; Mutations; Permutations; Sorting; Super oriented cycles; Transpositions; Upper bound.