An H(p, q)-labeling of a graph G is a vertex mapping f : VG → VH such that the distance (in the graph H) of f(u) and f(v) is at least p (at least q) if the vertices u and v are adjacent in G (are at distance two in G, respectively). This notion generalizes the notions of L(p, q)and C(p, q)-labelings of graphs studied as a graph model of the Frequency Assignment Problem. We study the computation...

In this article we study the treewidth of the display graph, an auxiliary graph structure obtained from the fusion of phylogenetic (i.e., evolutionary) trees at their leaves. Earlier work has shown that the treewidth of the display graph is bounded if the trees are in some formal sense topologically similar. Here we further expand upon this relationship. We analyse a number of reduction rules w...

In the previous lesson we talked about building evolution trees from sequences. Evolution trees represents the chronology of mutations during the evolution. The leaves are the observed species, and the inner nodes represents the later bifurcation during the evolution. Our target is to build those trees according to the sequences: (sequences =⇒ trees). One way to do that, which we have showed in...

We present two algorithms for computing the quartet distance between trees of arbitrary degree. The quartet distance between two unrooted evolutionary trees is the number of quartets—sub-trees induced by four leaves—that differs between the trees. Previous algorithms focus on computing the quartet distance between binary trees. In this paper, we present two algorithms for computing the quartet ...

The PQ-tree is a fundamental data structure that can encode large sets of permutations. It has recently been used in comparative genomics to model ancestral genomes with some uncertainty: given a phylogeny for some species, extant genomes are represented by permutations on the leaves of the tree, and each internal node in the phylogenetic tree represents an extinct ancestral genome, represented...

We present two algorithms for computing the quartet distance between trees of arbitrary degree. The quartet distance between two unrooted evolutionary trees is the number of quartets—sub-trees induced by four leaves—that differs between the trees. Previous algorithms focus on computing the quartet distance between binary trees. In this paper, we present two algorithms for computing the quartet ...

Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Recently, a new distance measure has been proposed: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. H...

Restricted rotation distance between pairs of rooted binary trees measures differences in tree shape and is related to rotation distance. In restricted rotation distance, the rotations used to transform the trees are allowed to be only of two types. Restricted rotation distance is larger than rotation distance, since there are only two permissible locations to rotate, but is much easier to comp...

An important problem in geometric computing is defining and computing similarity between two geometric shapes, e.g. point sets, curves and surfaces, etc. Important geometric and topological information of many shapes can be captured by defining a tree structure on them (e.g. medial axis and contour trees). Hence, it is natural to study the problem of comparing similarity between trees. We study...

Heavy metal pollutionis one of the largest problems in the Environment and human being. In industrial and urban areas trees can give better quantifications for pollutant concentrations and atmospheric deposition than non-biological samples. In order to know ability of Fraxinus excelsior­ (Ash tree) to reduce the concentrations of heavy metals and compared with the concentration of thes...