Domination integrity and efficient fuzzy graphs

In this paper, domination integrity of fuzzy graph and efficient fuzzy graph concepts is introduced with examples. An algorithm is developed to find whether an arc is strong or not. If it is strong, another algorithm will classify it as alpha strong arc and beta strong arc. The next algorithm is used to find whether the given fuzzy graph is a fuzzy tree or not. Domination and integrity are two different parameters used to define the stability of a graph in various situations. Using the strong arc concept a new parameter, domination integrity is defined and lower and upper bounds are found. This paper discusses the domination integrity for standard graphs such as path, cycle and complete graph. The domination integrity for Cartesian product of fuzzy graphs is also discussed. Finally, the new class of fuzzy graph, efficient fuzzy graph, is introduced. Efficient fuzzy graph is a special type of fuzzy graph that has the same dominating set, other than vertex set V, for both fuzzy graph and its underlying crisp graph.