Augmenting forests to meet odd diameter requirements

Given a graph G=(V,E) and an integer D≥1, we consider the problem of augmenting G by the smallest number of new edges so that the diameter becomes at most D. It is known that no constant approximation algorithms to this problem with an arbitrary graph G can be obtained unless P=NP. For a forest G and an odd D≥3, it was open whether the problem is approximable within a constant factor. In this paper, we give the first constant factor approximation algorithm to the problem with a forest G and an odd D; our algorithm delivers an 8-approximate solution in O(|V|³) time. We also show that a 4-approximate solution to the problem with a forest G and an odd D can be obtained in linear time if the augmented graph is additionally required to be biconnected.     

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ¹ (resp., ) the minimal Martin boundary ofW (resp., ). For ζ ∈ Δ, let ζ be the (cardinal) number of the set of pionts which lie over ζ and the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following: , where is the number of components of π^-1 M and π is the projection of ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.     

Hydrogen-bonded extended structure of the 1:3 adduct of a C3 symmetric cobalt(III) complex with a tripod-ligand involving three imidazolate groups and hydroquinone or resorcinol

The assembly reaction arising from hydrogen bonding between a chiral C3 symmetric cobalt(III) complex and a tripod-ligand involving three imidazolate groups [tris[2-(((2-methylimidazolato-4-yl)methylidene)amino)ethyl]amine]cobalt(III) and either hydroquinone or resorcinol gave the 1:3 adducts, with 3D extended structures showing the template effect of the complex.     

Conformational Control of a Polyether-Linked Porphyrin Dimer Induced by Complexation with a Sodium Cation

Abstract

A porphyrin dimer with a polyether linkage was newly synthesized and its conformational change triggered by complexation with an sodium cation was examined. It was confirmed by ¹H NMR titration experiments that the dimer bearing a pentaoxyehtylene linkage (1b) was effectively formed a 1:1 complex with a sodium perchlorate. A detailed investigation of the CIS values for the protons confirmed that the complex of 1b?Na⁺ adopts a slipped face-to-face conformation.