Year: 2025
Type: article
Abstract: A dominating set S ⊆ V is a co-secure dominating set if for each u ∈ S there exists v ∈ V S such that v is adjacent to u and (S {u}) ∪ {v} is a dominating set. The cardinality of a minimum co-secure dominating set in G is called the cosecure domination number of G and is denoted by γ_cs(G). The study of a co-secure dominating set is important in interconnection networks as it studies its security. In cosecure domination, a guard can ensure the safety of only one of its adjacent unguarded vertices. This motivated us to define a new domination parameter called complete co-secure domination, in which a guard can move to any one of its adjacent unguarded vertices without compromising the protection of G. A co-secure dominating set S is called a complete co-secure dominating set if for every u ∈ S and for every v ∈ V S that is adjacent to u, (S {u})∪ {v} is a dominating set. The cardinality of a minimum complete co-secure dominating set is called the complete co-secure domination number of G and is denoted by γ_ccs(G). In this paper, we study the complete co-secure domination in graphs and determined the lower and upper bounds and have checked their sharpness. We have proved that for any positive integer m, there exists a graph whose co-secure domination number is m and complete co-secure domination number is b, where m ≤ b ≤ 2m. We characterize graphs G such that γ_cs(G) = γ_ccs(G). We obtain a condition for which γ_cs(G) = γ_ccs(G) = γ_s(G) for graphs with δ(G) ≥ 2, thus partially resolving a question posed in paper from Arumugam, Ebadi and Manrique from 2014. We also obtain the complete co-secure domination number of some families of graphs.
Authors: Saraswathy, Gisha, Menon, Manju K.
Institution: St. Paul's College, Kalamassery
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7140151548
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