International Journal of Mathematics Trends and Technology
Volume 66 | Issue 2 | Year 2020 | Article Id. IJMTT-V66I2P506 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I2P506
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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Enrico L. Enriquez, "Fair Secure Domination in Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 2, pp. 49-57, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I2P506
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