Study on nonbondage number, connected domination number and toral global domination number of graphs

Abstract

A set D of vertices in a graph G = (V, E) is a dominating set of (j if every vertex in V-D is adjacent to some vertex in D. The domination number of G i, the minimum cardinality taken over all minimal dominating sets in G and is denoted by y(G). The nonbondage number of a graph G is the maximum cardinality among all sets of edges X <;; E(G) such iliat y(G~X)=y(G) and it is denoted by bll(G).IIIth~ same way, we call define total dominalJon number, eonnect~d domination number, global domination number and total global domination number of graphs. DilTerent types of methods arc available depending on types of the problems. Some exact values for the nonhondage number of graphs arc found. Upper bounds are ohtained for nonbondage number of a graph and the exact values ure determined for s~v~ral c1asse<;of graphs. \Ve have illustrated with examples some various results for the connected domillation number of graphs of standard graphs with better explanation.The exact values of connected domination nnmber and global domination number for somc standard graphs are calculated with the help of methods used by Kulli, Shampathkumar, Janakiram etc, We hav~ abo establi,hed some theorems relaled with the total global domination number of graph" In order to minimize the direct communication links among the transmitting stations under communication networks where maximum number of links that should be dropped to aeeompli,h this task is the nonboodage of a graph. In th~ similar way we can also apply eonnect~d domination number and total global domination number in various 'ways.