The Harary index H can be viewed as a molecular structure descriptor composed of increments representing interactions between pairs of atoms, such that their magnitude decreases with the increasing distance between the respective two atoms. A generalization of the Harary index, denoted by Hk, is achieved by employing the Steiner-type distance between k-tuples of atoms. We show that the linear c...

For a graph G = ( V , E ) and set S ⊆ of size at least 2, an S-Steiner tree T is subgraph that with . Two -Steiner trees ′ are internally disjoint (resp. edge-disjoint if ∩ ∅ ). Let κ λ denote the maximum number edge-disjoint) in The k-tree connectivity k edge-connectivity then defined as minimum ), where ranges over all -subsets In Li et al. (2018) [12] authors conjectured connected has vertic...

A combinatorial characterization of resolvable Steiner 2-(v, k, 1) designs embeddable as maximal arcs in a projective plane of order (v − k)/(k − 1) is proved, and a generalization of a conjecture by Andries Brouwer [9] is formulated.

We consider the k-Directed Steiner Forest (k-DSF) problem: given a directed graph G = (V, E) with edge costs, a collection D ⊆ V × V of ordered node pairs, and an integer k ≤ |D|, find a minimum cost subgraph H of G that contains an st-path for (at least) k pairs (s, t) ∈ D. When k = |D|, we get the Directed Steiner Forest (DSF) problem. The best known approximation ratios for these problems ar...

Given a set P of terminals in the plane and a partition of P into k subsets P1, . . . , Pk, a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree Ti connecting the terminals in each set Pi (i = 1, . . . , k) and a top-level tree Ttop connecting the trees T1, . . . , Tk. The goal is to minimize the total length of all trees. This problem arises naturally in the design of lo...

A singular direct product construction is presented for Steiner quadruple systems with a spanning block design. More constructions are also provided using Steiner systems S(3; k; v) and other designs. Small orders for v = 40 and 52 are constructed directly. Some in1nite classes of orders are also obtained. c © 2002 Elsevier Science B.V. All rights reserved.

On Approximate Min-Max Theorems for Graph Connectivity Problems Lap Chi Lau Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2006 Given an undirected graph G and a subset of vertices S ⊆ V (G), we call the vertices in S the terminal vertices and the vertices in V (G) − S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high...