Exponential domination and bondage numbers in some graceful cyclic structure

The domination number is an important vulnerability parameter that it has become one of the most widely studied topics in graph theory, and also the bondage number which is related by domination number the most often studied property of vulnerability of communication networks. Recently, Dankelmann et al. defined the exponential domination number denoted by ?e(G) in [17]. In 2016, the exponential bondage number, denoted by bexp(G), is defined by bexp(G) = min{|Be| : Be ? E(G), ?e(G - Be) > ?e(G)}, where ?e(G) is the exponential domination number of G [24]. In this paper, the above mentioned parameters is has been examined. Then exact formulas are obtained for the families of cyclic structures tend to have graceful subfamilies such as helm graph, windmill graph, circular necklace and friendship graph. © 2017 InforMath Publishing Group.