Approximation Algorithms for Broadcasting in Simple Graphs with Intersecting Cycles

Broadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes by placing a series of calls along the communication lines of the network. Every time the informed nodes aid the originator in distributing the message. Finding the minimum broadcast time of any vertex in an arbitrary graph is NP-Complete. The problem remains NP-Complete even for planar graphs of degree 3 and for a graph whose vertex set can be partitioned into a clique and an independent set. The best theoretical upper bound gives logarithmic approximation. It has been shown that the broadcasting problem is NP-Hard to approximate within a factor of 3-ɛ. The polynomial time solvability is shown only for tree-like graphs; trees, unicyclic graphs, tree of cycles, necklace graphs and some graphs where the underlying graph is a clique; such as fully connected trees and tree of cliques. In this thesis we study the broadcast problem in different classes of graphs where cycles intersect in at least one vertex. First we consider broadcasting in a simple graph where several cycles have common paths and two intersecting vertices, called a k-path graph. We present a constant approximation algorithm to find the broadcast time of an arbitrary k-path graph. We also study the broadcast problem in a simple cactus graph called k-cycle graph where several cycles of arbitrary lengths are connected by a central vertex on one end. We design a constant approximation algorithm to find the broadcast time of an arbitrary k-cycle graph. Next we study the broadcast problem in a hypercube of trees for which we present a 2-approximation algorithm for any originator. We provide a linear algorithm to find the broadcast time in hypercube of trees with one tree. We extend the result for any arbitrary graph whose nodes contain trees and design a linear time constant approximation algorithm where the broadcast scheme in the arbitrary graph is already known. In Chapter 6 we study broadcasting in Harary graph for which we present an additive approximation which gives 2-approximation in the worst case to find the broadcast time in an arbitrary Harary graph. Next for even values of n, we introduce a new graph, called modified-Harary graph and present a 1-additive approximation algorithm to find the broadcast time. We also show that a modified-Harary graph is a broadcast graph when k is logarithmic of n. Finally we consider a diameter broadcast problem where we obtain a lower bound on the broadcast time of the graph which has at least (d+k-1 choose d) + 1 vertices that are at a distance d from the originator, where k >= 1

Improving the Approximation Ratio of the Maximum Agreement Forest (MAF) on k trees and Estimating the Approximation Ratio of the Acyclic-MAF on k trees

Molecular phylogenetics has long been a well-established field of scientific research where the structure of the phylogenetic tree has been analysed to know about the evolutionary process of the organism. In biology, leaf-labelled trees are widely used to describe the evolutionary relationships. In this setting, the leaves of the tree correspond to extant species, and the internal vertices represent the ancestral species. However, for certain species, evolution is not completely tree-like. Reticulation events such as horizontal gene transfer (HGT), hybridization and recombination play a significant role in the evolution of the species. Suppose we have two phylogenetic trees each of which is for a gene of the same set of species. Due to reticulate evolution the two gene trees, though related, appear different. As a result, instead of the tree like structure, a phylogenetic network is widely viewed as a most suitable tool to represent reticulation. A phylogenetic network contains hybrid nodes for the species evolved from two parents. The distance between two phylogenetic trees can be computed with the help of a Maximum Agreement Forest (MAF) of those trees. The fewer components in MAF, the greater is the similarity between the two trees. This number of components in that agreement forest shows how many edges from each of the two trees need to be cut so that the resulting forest agree after all forced edge contractions. Recent research reveals that the MAF on k trees can be approximated within a ratio of 8. We have given a better approximation ratio for the MAF on k trees and also provide an approximation ratio for Maximum Acyclic Agreement Forest (MAAF) on k (\u3e=2) trees

(7aR*,12bS*)-8,12b-Dihydro-7aH-indeno­[1′,2′:5,6][1,4]selenazino[2,3,4-ij]quinolin-13-ium hydrogen sulfate

In the title compound, C18H14NSe+·HSO4 −, the cyclo­pentene ring in the cation has an envelope conformation while the central six-membered 1,4-selenazine ring adopts a sofa conformation. The dihedral angle between the planes of the terminal benzene rings is 68.08 (11)°. In the crystal, the anions form chains along the c axis through O—H⋯O hydrogen bonds. Weak C—H⋯O and C—H⋯π hydrogen bonds, as well as attractive Se⋯Se [3.5608 (8) Å] inter­actions, further consolidate the crystal structure

Synthesis and characterization of some gold(I)-thiolate complexes having N-methylimidazole moiety

1303-1307Antithyroid drugs inhibit the thyroid hormone synthesis by inactivating the thyroid peroxidase and/or iodothyronine deiodinase, which are involved in iodination and deiodination reactions. Gold(I) compounds also inhibit the thyroid hormone synthesis by interacting with the selenocysteine residue of iodothyronine deiodinase. However, the chemical reactions between these two different classes of compounds have not been studied. In this paper, we describe the interaction of therapeutic gold(I) compounds with the commonly used thiourea-based antithyroid drug, methimazole. It is observed that the gold(I) phosphine complexes (R3PAuCl, where R = Me, Et, Ph) react with methimazole only upon deprotonation to produce the corresponding gold(I)-thiolate complexes. Addition of PPh3 to the gold(I)-thiolates produces (R3PAuPPh3)+ (R = Me or Et), indicating the possibility of ligand exchange reactions

Antithyroid drugs and their analogues protect against peroxynitrite-mediated protein tyrosine nitration - a mechanistic study

In this paper, the effect of some commonly used antithyroid drugs and their analogues on peroxynitrite-mediated nitration of proteins is described. The nitration of tyrosine residues in bovine serum albumin (BSA) and cytochrome c was studied by Western blot analysis. These studies reveal that the antithyroid drugs methimazole (MMI), 6-n-propyl-2-thiouracil (PTU), and 6-methyl-2-thiouracil (MTU), which contain thione moieties, significantly reduce the tyrosine nitration of both BSA and cytochrome c. While MMI exhibits good peroxynitrite (PN) scavenging activity, the thiouracil compounds PTU and MTU are slightly less effective than MMI. The S- and Se- methylated compounds show a weak inhibitory effect in the nitration of tyrosine, indicating that the presence of a thione or selone moiety is important for an efficient inhibition. Similarly, the replacement of N—H moiety in MMI by N-methyl or N-m-methoxybenzyl substituents dramatically reduces the antioxidant activity of the parent compound. Theoretical studies indicate that the substitution of N—H moiety by N—Me significantly increases the energy required for the oxidation of sulfur center by PN. However, such substitution in the selenium analogue of MMI increases the activity of parent compound. This is due to the facile oxidation of the selone moiety to the corresponding selenenic and seleninic acids. Unlike N,N'-disubstituted thiones, the corresponding selones efficiently scavenge PN, as they predominantly exist in their zwitterionic forms in which the selenium atom carries a large negative charge

Gold(I)-selenolate complexes: Synthesis, characterization and ligand exchange reactions

Abstract. In this paper, the synthesis and characterization of some imidazole-based gold-selenolates are described. This study indicates that the nature of selenolate plays an important role in ligand exchange reactions in gold(I) selenolates. Furthermore, the reactivity of imidazole-based gold(I) selenolates toward nucleophiles such as selenols and phosphines is strikingly different from that of the N,N-dimethylaminobenzylamine-based gold(I) complexes. The presence of Se· · · N non-bonded interactions in N,N-dimethylaminobenzylamine-based gold(I) complexes modulates the reactivity of Au(I) centre towards incoming nucleophiles

Amide-based glutathione peroxidase mimics: effect of secondary and tertiary amide substituents on antioxidant activity

Sec or terts: A series of sec- and tert-amide substituted diselenides have been synthesised as synthetic mimics of glutathione peroxidase (GPx), characterized, and studied for their antioxidant activities using H2O2, Cum-OOH, and tBuOOH as substrates and PhSH as thiol co-substrate. The substitution at the free NH group of the amide moiety in the sec-amide based diselenides is shown to enhance the GPx activity. A series of secondary and tertiary amide-substituted diselenides were synthesized and studied for their GPx-like antioxidant activities using H2O2, Cum-OOH, and tBuOOH as substrates and PhSH as thiol co-substrate. The effect of substitution at the free -NH group of the amide moiety in the sec-amide-based diselenides on GPx activity was analyzed by detailed experimental and theoretical methods. It is observed that substitution at the free -NH group significantly enhances the GPx-like activities of the sec-amide-based diselenides, mainly by reducing the Se···O nonbonded interactions. The reduction in strength of the Se···O interaction upon introduction of N,N-dialkyl substituents not only prevents the undesired thiol exchange reactions, but also reduces the stability of selenenyl sulfide intermediates. This leads to a facile disproportionation of the selenenyl sulfide to the corresponding diselenide, which enhances the catalytic activity. The mechanistic investigations indicate that the reactivity of diselenides having sec- or tert-amide moieties with PhSH is extremely slow; indicating that the first step of the catalytic cycle involves the reaction between the diselenides and peroxide to produce the corresponding selenenic and seleninic acids