Disjoint Perfect Secure Domination in the Join and Corona of Graphs
Renelyn B. Udtohan -, Enrico L. Enriquez Β· International Journal of Latest Engineering Research and Applications (IJLERA) Β· 2024
Let πΊπΊ be a graph.A dominating set π·π· β ππ(πΊπΊ)is called a secure dominating set of πΊπΊ if for each vertex π’π’ β ππ(πΊπΊ) β π·π·, there exists a vertex π£π£ β π·π· such that π’π’π£π£ β πΈπΈ(πΊπΊ) and the set (π·π· β {π£π£}) βͺ {π’π’} is a dominating set of πΊπΊ.If every π’π’ β ππ(πΊπΊ) β π·π· is adjacent to exactly one vertex in π·π·, then π·π· is a perfect secure dominating set of πΊπΊ.Let π·π· be a minimum perfect secure dominating set of G.If ππ β ππ(πΊπΊ) β π·π· is a perfect secure dominating set of πΊπΊ, then ππ is called an inverse perfect secure dominating set of πΊπΊ with respect to π·π·.A disjoint perfect secure dominating set of πΊπΊ is the set πΆπΆ = π·π· βͺ ππ β ππ(πΊπΊ).Furthermore, the disjoint perfect secure domination number, denoted by πΎπΎ ππππ πΎπΎ ππππ (πΊπΊ), is the minimum cardinality of a disjoint perfect secure dominating set of πΊπΊ.A disjoint perfect secure dominating set of cardinality πΎπΎ ππππ πΎπΎ ππππ (πΊπΊ) is called πΎπΎ ππππ πΎπΎ ππππ -set.In this paper, we initiate a study of the concept of disjoint perfect secure domination in graphs and characterize this type of domination in graphs under some binary operations, namely the join and corona of two graphs.