Different types of degrees of a node in fuzzy graphs are discussed. Bounds for the domination number of fuzzy graphs are obtained. Domination in fuzzy trees is studied. The concept of strong domination number of fuzzy graphs is introduced by using membership values of strong arcs in fuzzy graphs. The strong domination number of complete fuzzy graph and complete bipartite fuzzy graph is determined and bounds for the strong domination number of fuzzy graphs are obtained. Also the relationship between the strong domination number of a fuzzy graph and that of its complement is discussed. The effect of the removal of a node or an arc on the minimum strong dominating set of a fuzzy graph is studied. With an illustrative example, the advantage of strong domination is explained. The concept of strong total domination in fuzzy graphs is introduced. The strong total domination number for several classes of fuzzy graphs is determined. A lower bound and an upper bound for the strong total domination number in terms of strong domination number are obtained. Strong total domination in fuzzy trees is studied. The concept of connected domination in fuzzy graphs using strong arcs is introduced.

Manjusha O. T. is an Assistant Professor of Mathematics at Kerala Government Polytechnic College, Kozhikode, Kerala, India. The author of many impact factored journal publications on strong domination in fuzzy graphs. She received the Ph.D degree (2017) in Mathematics from National Institute of Technology (NIT), Calicut, Kerala, India.