Fair Secure Domination in Graphs

 International Journal of Mathematics Trends and Technology (IJMTT) © 2020 by IJMTT Journal Volume-66 Issue-2 Year of Publication : 2020 Authors : Enrico L. Enriquez 10.14445/22315373/IJMTT-V66I2P506

MLA Style:Enrico L. Enriquez  "Fair Secure Domination in Graphs" International Journal of Mathematics Trends and Technology 66.2 (2020):49-57.

APA Style: Enrico L. Enriquez (2020). Fair Secure Domination in Graphs International Journal of Mathematics Trends and Technology, 49-57.

Abstract
Let G be a connected simple graph. A dominating set S ⊆ V(G) is a fair dominating set in G if for ever two distinct vertices u and v from V(G)\S, \N(u) ∩ S| = |N(v) ∩ S|, that is, every two distinct vertices not in S have the same number of neighbors from S. A fair dominating set S ⊆ V(G) is a fair secure dominating set if for each u ε V(G)\S, there exists V ε S such that uv ε E(G) and the set (S\{v}) U {u} is a dominating set of G. The minimum cardinality of a fair secure dominating set of G, denoted by γfsd(G), is called the fair secure domination number of G. In this paper, we initiate the study of the concept and give some realization problems. In particular, we show that given positive integers k,m, and n≥2 such that 1≤ k ≤ m ≤ n-1, there exists a connected nontrivial graph G with |V(G)|=n such that γfd(G) = k and γfsd(G) =m. Further, we show the characterization of the fair secure dominating set in the join of two nontrivial connected graphs.

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Keywords
dominating set, fair dominating set, secure dominating set, fair secure dominating set