Fair Restrained Domination in Graphs

Enrico L. Enriquez

doi:https://doi.org/10.14445/22315373/IJMTT-V66I1P530

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 1 | Year 2020 | Article Id. IJMTT-V66I1P530 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I1P530

Fair Restrained Domination in Graphs

Enrico L. Enriquez

Citation :

Enrico L. Enriquez, "Fair Restrained Domination in Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 1, pp. 229-235, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I1P530

Abstract

Let G be a connected simple graph. A dominating set S ⊆ V(G) is a fair dominating set in G if every two distinct vertices not in S have the same number of neighbors from S, that is, for every two distinct vertices U and V from V(G) \ S, |N(u)∩ S| = |N(v) ∩ S|. A fair dominating set S ⊆V(G) is a fair restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V(G)\S. Alternately, a fair dominating set S⊆ V(G) is a fair restrained dominating set if N[S] = V(G) and < V(G)\ S> is a subgraph without isolated vertices. The minimum cardinality of a fair restrained dominating set of G, denoted by γfrd(G), is called the fair restrained 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 ≥3 such the 1≤ k≤ m ≤n-2, there exists a connected nontrivial graph G with |V(G)| = n such that γfd(G) = k and γfrd(G) = m. Further, we show the characterization of the fair restrained dominating set in the join of two nontrivial connected graphs.

Keywords

dominating set, fair dominating set, restrained dominating set, fair restrained dominating set

References

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