Genome Rearrangements on Multigenomic Models: Applications of Graph Convexity Problems

Genome rearrangements are events where large blocks of DNA exchange pieces during evolution. The analysis of such events is a tool for understanding evolutionary genomics, in whose context many rearrangement distances have been proposed, based on finding the minimum number of rearrangements to transform one genome into another, using some predefined operation. However, when more than two genomes are considered, we have new challenging problems. Studying such problems from a combinatorial point of view has been shown to be a useful tool to approach such problems, for example, the reconstruction of phylogenetic trees. We focus on genome rearrangement problems related to graph convexity. Such an approach is in connection with some other well-known studies on multigenomic models, for example, those based on the median and on the closest string. We propose an association between graph convexities and genome rearrangements in such a way that graph convexity problems deal with input sets of vertices and try to answer questions concerning the closure of such inputs. The concept of closure is useful for studies on genome rearrangement by suggesting mechanisms to reduce the genomic search space. Regarding the computational complexity, and considering the Hamming distance on strings, we solve the following problems: decide if a given set is convex; compute the interval and the convex hull of a given set; and determine the convexity number, interval number, and hull number of a Hamming graph. All such problems are solved for three types of convexities: geodetic, monophonic, and P₃. Considering the Cayley distance on permutations, we solve the convexity number and interval determination problems for the geodetic convexity.

Computational aspects of the Helly property: a survey

In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property. Say that a family of subsets has the Helly property when every subfamily of it, formed by pairwise intersecting subsets, contains a common element. There are many generalizations of this property which are relevant to some parts of mathematics and several applications in computer science. In this work, we survey computational aspects of the Helly property. The main focus is algorithmic. That is, we describe algorithms for solving different problems arising from the basic Helly property. We also discuss the complexity of these problems, some of them leading to NP-hardness results.