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.
