Picture fuzzy graph is the generalization of fuzzy graph and intuitionistic fuzzy graph. PFGs provide a more flexible and expressive way to represent uncertainty and vagueness in graph-based data, compared to traditional fuzzy graphs. Total domination is an important topic for its entire domination characteristics. In this paper, some domination parameters like complementary nil domination number, inverse domination number and total dominating set, total domination number are introduced in a picture-fuzzy environment. We have introduced picture fuzzy vertex cardinality, picture fuzzy edge cardinality, cardinality of a PFG. Studied their nature with domination parameters. A novel method to get a join of two PFGs has been discussed. A new definition of complement of a PFG is presented. Few findings regarding the complementary nil dominating set, complementary nil domination number in bipartite PFG, complete bipartite PFG are developed. The concept of enclave is introduced in PFG and proved some relations between enclave and complementary nil dominating set. Also, total domination number, and the independent domination number of complete, complement, and join of PFGs have been presented. We presented some theorems associated with them. We proved that domination, independent domination and total domination number are equal for a complete PFG. Finally, we demonstrate an application of the total domination concept to identify the most critical junction in a railway network.
Banerjee et al. (Fri,) studied this question.
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