Study on domination and total domination of graphs

Abstract

A set D of vertices in a graph G -=0 (V,E) is a dominating set of G if every vertex in V - D is adjacent to some vertex in D. The domination number of G is the minimum cardinality taken over all minimal dominating sets in G and is denoted by reG). For a graph G with reG) > 1, the cobondage number cb(G)of G is defined by cb(G)=min{IE~I:Eo cE(G) and y(G+Eo)<r(G)}. The bondage number beG) of a graph G is the minimum cardinality of a set of edges of G whose removal from G results in a graph with domination number larger than that of G. In the same way we can define two cobondage number, total cobondage number, two bondage number and total bondage number. Different types of methods are available depending on types of problems. Sharp upper bOWlds are obtained for cobondage number of a graph and the exact values are deterrni~ed for several classes of graphs. The exact values of total cobondage number for some standard graphs are calculated with the help of the methods used by Cockaync, Hedetniemi, Hartnell, Rail, Kulli, etc. An alternative proof of a theorem for total bondage number of Kulli for a complete graph with at least five vertices 1S provided. Finally, some operations on two bondage number are developed.