Domination and Independence in Fuzzy Semigraphs

Objectives: To identify dominating and independent sets in fuzzy semigraphs by developing suitable methods. To examine dominating and independent sets in fuzzy semigraphs for their characteristics and behaviors. The goal is to examine the use of the concepts of domination and independence in the fields of analysis of social networks, security of networks, and allocation of resources. Methods: This study introduces the concept of strong arc fuzzy semigraph and explores various parameters such as dominating and independent sets. The Minimum Adjacent Dominating Number (ad-number) γ_a(𝓖) and Maximum Adjacent Dominating Number (ad-number) Γ_a(𝓖) , Minimum End Node Adjacent Dominating Number (ead-number) γₑₐ(𝓖) and Maximum End Node Adjacent Dominating Number (ead-number) Γₐₑ(𝓖) as well as Minimum Consecutive Adjacent Dominating Number (cad-number) γca(𝓖) and Maximum Consecutive Adjacent Dominating Number (cad-number) Γca(𝓖) . These parameters represent the respective minimum and maximum cardinalities taken over all minimal dominating sets of nodes of 𝓖 . Additionally, it explores the e-independence dominating number i_e and e-independence number β_e(𝓖) , strongly independence dominating number i_s and strongly independence number β_s(𝓖) and consecutive adjacent independence dominating number ica and consecutive adjacent independence number βca(𝓖) . These parameters represent the respective minimum and maximum cardinalities derived from all maximal independent sets of nodes in 𝓖 . Findings: The study develops a thorough correlation between fuzzy dominant sets and independent sets in the fuzzy semigraph 𝓖 . The statement illustrates the way in which the introduced factors interact and impact the structure and behaviour of the fuzzy semigraph. Novelty: The introduction of strong arc fuzzy semigraphs is a novel notion in the examination of fuzzy graphs. The study involves the creation and examination of novel parameters associated with dominant and independent sets in fuzzy semigraphs. This study aims to establish novel theoretical linkages between fuzzy domination and independent sets, therefore expanding our knowledge of the structural features of fuzzy semigraphs.