| Summary: | In this paper, the concept of a strong <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-Connected Total Perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set and a weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set in fuzzy graphs is introduced. In the current work, the triple-connected total perfect dominating set is modified to an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula>(G) and number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>D</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> New definitions are compared with old ones. Strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set and number of fuzzy graphs are obtained. The results of those fuzzy sets are discussed with the definitions of spanning fuzzy graphs, strong and weak arcs, dominating sets, perfect dominating sets, generalization of triple-connected total perfect dominating sets of fuzzy graphs, complete, connected, bipartite, cut node, tree, bridge and some other new notions of fuzzy graphs which are analyzed with a strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula>(G) set of fuzzy graphs. The order and size of the strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="normal">G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> fuzzy set are studied. Additionally, a few related theorems and statements are analyzed.
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