A generic greedy algorithm, partially-ordered graphs and NP-completeness.

Let π be any fixed polynomial-time testable, non-trivial, hereditary property of graphs. Suppose that the vertices of a graph G are not necessarily linearly ordered but partially ordered, where we think of this partial order as a collection of (possibly exponentially many) linear orders in the natural way. We prove that the problem of deciding whether a lexicographically first maximal subgraph of G satisfying π, with respect to one of these linear orders, contains a specified vertex is NP-complete

Greedy algorithms, H-colourings and a complexity-theoretic dichotomy

Let H be a fixed undirected graph. An H-colouring of an undirected graph G is a homomorphism from G to H. If the vertices of G are partially ordered then there is a generic non-deterministic greedy algorithm which computes all lexicographically first maximal H-colourable subgraphs of G. We show that the complexity of deciding whether a given vertex of G is in a lexicographically first maximal H-colourable subgraph of G is NP-complete, if H is bipartite, and ${\bf \Sigma}_2^p$-complete, if H is non-bipartite. This result complements Hell and Nesetril's seminal dichotomy result that the standard H-colouring problem is in P, if H is bipartite, and NP-complete, if H is non-bipartite. Our proofs use the basic techniques established by Hell and Nesetril, combinatorially adapted to our scenario

A generic greedy algorithm, partially-ordered graphs and NP-completeness

Let π be any fixed polynomial-time testable, non-trivial, hereditary property of graphs. Suppose that the vertices of a graph G are not necessarily linearly ordered but partially ordered, where we think of this partial order as a collection of (possibly exponentially many) linear orders in the natural way. We prove that the problem of deciding whether a lexicographically first maximal subgraph of G satisfying π, with respect to one of these linear orders, contains a specified vertex is NP-complete

Drosophila melanogaster as a Model to Study the Phenotypes related to Genome Stability of the Fragile-X syndrome.

Fragile-X syndrome is one of the most common forms of inherited mental retardation and autistic behaviors. The reduction/absence of the functional FMRP protein, coded by the X-linked Fmr1 gene in humans, is responsible for the syndrome. Patients exhibit a variety of symptoms predominantly linked to the function of FMRP protein in the nervous system like autistic behavior and mild-to-severe intellectual disability. Fragile-X (FraX) individuals also display cellular and morphological traits including branched dendritic spines, large ears, and macroorchidism. The dFmr1 gene is the Drosophila ortholog of the human Fmr1 gene. dFmr1 mutant flies exhibit synaptic abnormalities, behavioral defects as well as an altered germline development, resembling the phenotypes observed in FraX patients. Therefore, Drosophila melanogaster is considered a good model to study the physiopathological mechanisms underlying the Fragile-X syndrome. In this review, we explore how the multifaceted roles of the FMRP protein have been addressed in the Drosophila model and how the gained knowledge may open novel perspectives for understanding the molecular defects causing the disease and for identifying novel therapeutical targets