Closure of genomic sets: applications of graph convexity to genome rearrangement problems

Abstract

Genome rearrangement problems aim at finding the minimum number of mutational events required to transform a genome into another. Studying such problems from a combinatorial point of view has been proved to be useful for the reconstruction of phylogenetic trees. In this work we focus on the relations between genome rearrangement and graph convexity in the following way. Graph convexity problems deal with input sets of vertices and try to understand properties on the closure of such inputs. In turn, the concept of closure is useful for studies on genome rearrangement by suggesting mechanisms to reduce the genomic search space. Computational complexity studies are widely developed in graph convexity problems, since there are several types of convexities, each capturing a distinct way of defining what is meant by “closure”. In this regard, considering the Hamming distance of strings, we solve the following problems: decide whether a given set is convex; compute the interval and the convex hull of a given set; and determine the convexity number of a given graph. All such problems are solved for the geodesic and monophonic convexities. On the other hand, for the Cayley distance of permutations, we study convexity of sets and interval determination of sets, in the context of the geodesic convexity.