International Journal of Mathematics Trends and Technology
Volume 66 | Issue 1 | Year 2020 | Article Id. IJMTT-V66I1P530 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I1P530
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.
[1] O. Ore., Theory of Graphs, American Mathematical Society, Provedence, R.I., 1962
[2] E.J. Cockayne, and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks, (1977) 247-261.
[3] N.A. Goles, E.L. Enriquez, C.M. Loquias, G.M. Estrada, R.C. Alota, z-Domination in Graphs, Journal of Global Research in Mathematical Archives, 5(11), 2018, pp 7-12.
[4] E.L. Enriquez, V.V. Fernandez, J.N. Ravina, Outer-clique Domination in the Corona and Cartesian Product of Graphs, Journal of Global Research in Mathematical Archives, 5(8), 2018, pp 1-7.
[5] E.L. Enriquez, G.M. Estrada, V.V. Fernandez, C.M. Loquias, A.D. Ngujo, Clique Doubly Connected Domination in the Corona and Cartesian Product of Graphs, Journal of Global Research in Mathematical Archives, 6(9), 2019, pp 1-5.
[6] E.L. Enriquez, E.S. Enriquez, Convex Secure Domination in the Join and Cartesian Product of Graphs, Journal of Global Research in Mathematical Archives, 6(5), 2019, pp 1-7.
[7] E.L. Enriquez, G.M. Estrada, C.M. Loquias, Weakly Convex Doubly Connected Domination in the Join and Corona of Graphs, Journal of Global Research in Mathematical Archives, 5(6), 2018, pp 1-6.
[8] J.A. Dayap, E.L. Enriquez, Outer-convex Domination in the Composition and Cartesian Product of Graphs, Journal of Global Research in Mathematical Archives, 6(3), 2019, pp 34-42.
[9] D.P. Salve, E.L. Enriquez, Inverse Perfect Domination in the Composition and Cartesian Product of Graphs, Global Journal of Pure and Applied Mathematics, 12(1), 2016, pp 1-10.
[10] E.L. Enriquez, and S.R. Canoy,Jr., Secure Convex Domination in a Graph, International Journal of Mathematical Analysis, Vol. 9, 2015, no. 7, 317-325.
[11] E.L. Enriquez, B.P. Fedellaga, C.M. Loquias, G.M. Estrada, M.L. Baterna, Super Connected Domination in Graphs, Journal of Global Research in Mathematical Archives, 6(8), 2019, pp 1-7.
[12] M.P. Baldado, Jr. and E.L. Enriquez, Super Secure Domination in Graphs, International Journal of Mathematical Archive- 8(12), 2017, pp. 145-149.
[13] R.T. Aunzo, Jr. and E.L. Enriquez, Convex Doubly Connected Domination in Graphs, Applied Mathematical Science, Vol. 9, 2015, no. 135, 6723-6734.
[14] E.L. Enriquez, Super Convex Dominating Set in the Corona of Graphs, International Journal of Latest Engineering Research and Applications, Vol. 4, 2019, no. 7, 11-16.
[15] G.M. Estrada, C.M. Loquias, E.L. Enriquez, and C.S. Baraca, Perfect Doubly Connected Domination in the Join and Corona of Graphs, International Journal of Latest Engineering Research and Applications, Vol. 4, 2019, no. 7, 17-21.
[16] J.L. Ranara, C.M. Loquias, G.M. Estrada, T.J. Punzalan, E.L. Enriquez, Identifying Code of Some Special Graphs, Journal of Global Research in Mathematical Archives, Vol. 5, 2018, no. 9, 1-8.
[17] Caro, Y., Hansberg, A.,Henning, M., Fair Domination in Graphs. University of Haifa, 1-7, 2011.
[18] E.L. Enriquez, Super Fair Dominating Set in Graphs, Journal of Global Research in Mathematical Archives, 6(2), 2019, pp 8-14.
[19] J.A. Telle, A. Proskurowski, Algorithms for Vertex Partitioning Problems on Partial-k Trees, SIAM J. Discrete Mathematics, 10(1997), 529-550.
[20] C.M. Loquias, and E.L. Enriquez, On Secure Convex and Restrained Convex Domination in Graphs, International Journal of Applied Engineering Research, Vol. 11, 2016, no. 7, 4707-4710.
[21] E.L. Enriquez, and S.R. Canoy,Jr., Restrained Convex Dominating Sets in the Corona and the Products of Graphs, Applied Mathematical Sciences, Vol. 9, 2015, no. 78, 3867-3873.
[22] E.L. Enriquez, Secure Restrained Convex Domination in Graphs, International Journal of Mathematical Archive, Vol. 8, 2017, no. 7, 1-5.
[23] E.L. Enriquez, On Restrained Clique Domination in Graphs, Journal of Global Research in Mathematical Archives, Vol. 4, 2017, no. 12, 73-77.
[24] E.L. Enriquez, Super Restrained Domination in the Corona of Graphs, International Journal of Latest Engineering Research and Applications, Vol. 3, 2018, no. 5, 1-6.
[25] E.M. Kiunisala, and E.L. Enriquez, Inverse Secure Restrained Domination in the Join and Corona of Graphs, International Journal of Applied Engineering Research, Vol. 11, 2016, no. 9, 6676-6679.
[26] T.J. Punzalan, and E.L. Enriquez, Inverse Restrained Domination in Graphs, Global Journal of Pure and Applied Mathematics, Vol. 3, 2016, pp 1-6.
[27] R.C. Alota, and E.L. Enriquez, On Disjoint Restrained Domination in Graphs, Global Journal of Pure and Applied Mathematics, Vol. 12, 2016, no. 3 pp 2385-2394.
[28] G. Chartrand and P. Zhang, A First Course in Graph Theory, Dover Publication, Inc., New York, 2012.
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
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