Some study on edge geodetic number of a graph

Abstract

The research work entitled quotSome Study on Edge Geodetic Number of a Graphquot, has been carried out under the guidance of Dr. Venkanagouda M Goudar, Professor, Department of Mathematics, Sri Siddhartha Institute of Technology, Tumkur, India. This thesis focuses on edge geodetic number of a graph. We obtain some new parameters on edge geodetic number of graphs. We have divided this thesis into six chapters.One concept that pervades all of graph theory is that of distance and distance is used in isomorphism testing, graph operations, hamiltonicity problems, enternal problems on connectivity and convexity in graphs. The distance d(u; v) between two vertices u and v of a graph G is the length of a shortest path joining them if any, otherwise d(u; v) = 1: This gives rise to the concept of geodetic set and geodetic number of graph. The geodetic concepts have applications in location theory and covexity theory.In this thesis, we define and develop various concepts viz, the split edge geodetic number, the total edge geodetic number, the doubly connected edge geodetic number and various operations on these concepts. These concepts have interesting application in location theory and covexity theory.Let G = (V;E) be a connected graph with atleast two vertices. A u and#1048576; v path of length d(u; v) is called a u and#1048576; v geodesic. We define I[u; v] to the set(interval) of all vertices lying on some u and#1048576; v geodesic of G and for a nonempty subset S of V (G), I[S] = [u;v2SI[u; v]. A set S of vertices of G is called a geodetic set in G if I[S] = V (G) and a geodetic set of minimum cardinality is a minimum geodetic set. The cardinality of a minimum geodetic set in G is called geodetic number of G and we denoted it by g(G).In Chapter 2, we introduce the concept of split edge geodetic number. It includes split edge geodetic number of standard graphs in terms of graph theoretic values. We also initiated to study the split edge geodetic number on the operations such as corona and cartesian product of two graphs.