For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steine...

Distance-balanced graphs are introduced as graphs in which every edge uv has the following property: the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Basic properties of these graphs are obtained. In this paper, we study the conditions under which some graph operations produce a distance-balanced graph.

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...

The Steiner connectivity problem is a generalization of the Steiner tree problem. It consists in finding a minimum cost set of simple paths to connect a subset of nodes in an undirected graph. We show that polyhedral and algorithmic results on the Steiner tree problem carry over to the Steiner connectivity problem; namely, the Steiner cut and the Steiner partition inequalities, as well as the a...

Perfect codes and optimal anticodes in the Grassman graph Gq(n, k) are examined. It is shown that the vertices of the Grassman graph cannot be partitioned into optimal anticodes, with a possible exception when n=2k. We further examine properties of diameter perfect codes in the graph. These codes are known to be similar to Steiner systems. We discuss the connection between these systems and ``r...

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

We consider the DIRECTED STEINER NETWORK problem, also called the POINT-TO-POINT CONNECTION problem, where given a directed graph G and p pairs {(s1, t1), . . . ,(sp, tp)} of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from si to ti for all i. The problem is NP-hard for general p, since the DIRECTED STEINER TREE problem is a special case. Until now, the ...

Given a weighted graph G = (V,E,w) with a set of k terminals T ⊂ V , the Steiner Point Removal problem seeks for a minor of the graph with vertex set T , such that the distance between every pair of terminals is preserved within a small multiplicative distortion. Kamma, Krauthgamer and Nguyen (SODA 2014, SICOMP 2015) used a ball-growing algorithm to show that the distortion is at most O(log k) ...

In this paper we prove that any distance-balanced graph $G$ with $Delta(G)geq |V(G)|-3$ is regular. Also we define notion of distance-balanced closure of a graph and we find distance-balanced closures of trees $T$ with $Delta(T)geq |V(T)|-3$.

In this paper, we discuss the problem of reoptimization of Steiner tree. We are given an instance of Graph and also an optimal Steiner tree of it. If some changes occurs later on in the given graph. New optimal Steiner tree is to be determine, this process is known as re optimization. We consider two cases of change: one is addition of a new edge and second is, Deletion of an existing edge. For...