Distances in and between graphs.

Thesis (M.Sc.)-University of Natal, 1991.Aspects of the fundamental concept of distance are investigated in this dissertation. Two major topics are discussed; the first considers metrics which give a measure of the extent to which two given graphs are removed from being isomorphic, while the second deals with Steiner distance in graphs which is a generalization of the standard definition of distance in graphs. Chapter 1 is an introduction to the chapters that follow. In Chapter 2, the edge slide and edge rotation distance metrics are defined. The edge slide distance gives a measure of distance between connected graphs of the same order and size, while the edge rotation distance gives a measure of distance between graphs of the same order and size. The edge slide and edge rotation distance graphs for a set S of graphs are defined and investigated. Chapter 3 deals with metrics which yield distances between graphs or certain classes of graphs which utilise the concept of greatest common subgraphs. Then follows a discussion on the effects of certain graph operations on some of the metrics discussed in Chapters 2 and 3. This chapter also considers bounds and relations between the metrics defined in Chapters 2 and 3 as well as a partial ordering of these metrics. Chapter 4 deals with Steiner distance in a graph. The Steiner distance in trees is studied separately from the Steiner distance in graphs in general. The concepts of eccentricity, radius, diameter, centre and periphery are generalised under Steiner distance. This final chapter closes with an algorithm which solves the Steiner problem and a Heuristic which approximates the solution to the Steiner problem

Aspects of distance measures in graphs.

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.In this thesis we investigate bounds on distance measures, namely, Steiner diameter and radius, in terms of other graph parameters. The thesis consists of four chapters. In Chapter 1, we define the most significant terms used throughout the thesis, provide an underlying motivation for our research and give background in relevant results. Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. In Chapter 2, we give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdös, Pach, Pollack and Tuza. We obtain improved bounds on diamn(G) for K3-free graphs and C4-free graphs. In Chapter 3, we prove that, if G is a 3-connected plane graph of order p and maximum face length l then the radius of G does not exceed p/6 + 5l/6 + 5/6. For constant l, our bound improves on a bound by Harant. Furthermore we extend these results to 4- and 5-connected planar graphs. Finally, we complete our study in Chapter 4 by providing an upper bound on diamn(G) for a maximal planar graph G

Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Steiner Distance in Product Networks

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.Comment: 29 pages, 4 figure