‎‎‎In this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $S$-posets‎, ‎posets with a monotone action of a‎ pomonoid $S$ on them‎, ‎is studied‎. ‎We give a criterion‎, ‎like the Baer condition for injectivity of modules‎, ‎or Skornjakov criterion for injectivity of $S$-sets‎, ‎for the order dense injectivity‎. ‎Also‎, ‎we consider such injectivit...

The problem of determining if a graph is 2-colourable (i.e., bipartite) has long been known to have a simple polynomial time algorithm. Being 2-colourable is equivalent to having a bipartition of the vertex set where each cell is &-free. We extend this notion to determining if there exists a bipartition where each cell is G-free for some fixed graph G. One might expect that for some graphs othe...

We present a method for summarising collections of phylogenetic trees that extends the notion of consensus trees. Each branch in a phylogenetic tree corresponds to a bipartition or split of the set of taxa labelling its leaves. Given a collection of phylogenetic trees, each labelled by the same set of taxa, all those splits that appear in more than a predefined threshold proportion of the trees...

In phylogenetics there are various methods available for understanding the evolutionary history of a set of species based on the analysis of its 4-element subsets. Guided by biological data, such techniques usually require the initial computation of a quartet-weight function, i.e., a function that assigns a weight to each bipartition of each 4-element subset into two parts of size two, from whi...

We consider the following problem: given a forest of gene family trees on a set of genomes, find a first speciation which splits these genomes into two subsets and minimizes the number of gene duplications that happened before this speciation. We call this problem the Minimum Duplication Bipartition Problem. Using a generalization of the Minimum Edge-Cut Problem, known as Submodular Function Mi...

We consider the following problem: given a set of gene family trees, spanning a given set of species, find a first speciation which splits these species into two subsets and minimizes the number of gene duplications that happened before this speciation. We call this problem the Minimum Duplication Bipartition Problem. Using a generalization of the Minimum Edge-Cut Problem, we propose a polynomi...

Horizontal gene transfer (HGT) is one of the main mechanisms driving the evolution of microorganisms. Its accurate identification is one of the major challenges posed by reticulate evolution. In this article, we describe a new polynomial-time algorithm for inferring HGT events and compare 3 existing and 1 new tree comparison indices in the context of HGT identification. The proposed algorithm c...

The direct product of graphs G = (V (G), E(G)) and H = (V (H), E(H)) is the graph, denoted as G×H, with vertex set V (G×H) = V (G)×V (H), where vertices (x1, y1) and (x2, y2) are adjacent in G × H if x1x2 ∈ E(G) and y1y2 ∈ E(H). Let n be odd and m even. We prove that every maximum independent set in Pn×G, respectively Cm×G, is of the form (A×C)∪(B× D), where C and D are nonadjacent in G, and A∪...