Volume 67 | Issue 12 | Year 2021 | Article Id. IJMTT-V67I12P506 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I12P506

Let G = (V(G), E(G)) be a connected simple graph. A subset S of V(G) is a dominating set of G if for every u ∈ V(G\S), there exists v ∈ S such that uv ∈ E(G). A dominating set D is called a restrained dominating set if for each u ∈ V(G)\D there exists v ∈ V(D) and z ∈ V(G)\D(z ≠ u) such that u is adjacent to v and z. Further, if D is a minimum restrained dominating set of G, then a restrained dominating set S ⊆ V(G)\D is called an inverse restrained dominating set of G with respect to D. A disjoint restrained dominating set of G is the set C = D ∪ S ⊆ V(G). In this paper, we investigate the concept and give some important results on disjoint restrained domination arising from the join and corona of two graphs.

[1] G. Chartrand and P. Zhang. A First Course in Graph Theory, Dover Publication, Inc., New York, 2012.

[2] Ore O., (1962), Theory of Graphs, American Mathematical Society, Provedence, R.I.

[3] Dayap, J.A. and Enriquez, E.L., (2020), Outer-convex domination in graphs, Discrete Mathematics, Algorithms and Applications, 12(01) (2020)
pp. 2050008., https://doi.org/10.1142/S1793830920500081

[4] Enriquez, E.L. and Ngujo, A.D., (2020), Clique doubly connected domination in the join and lexicographic product of graphs, Discrete
Mathematics, Algorithms and Applications, 12(05), pp. 2050066. , https://doi.org/10.1142/S1793830920500664

[5] Enriquez, E.L. and Canoy,Jr., S.R., (2015), On a Variant of Convex Domination in a Graph, International Journal of Mathematical Analysis, 9(32),
pp. 1585-1592.

[6] Enriquez, E.L. and Canoy,Jr. S.R., (2015), Secure Convex Domination in a Graph, International Journal of Mathematical Analysis, 9(7), pp. 317-
325.

[7] Gomez, L.P. and Enriquez, E.L., (2020), Fair Secure Dominating Set in the Corona of Graphs, International Journal of Engineering and
Management Research, 10(03), pp. 115-120 , https://doi.org/10.31033/ijemr.10.3.18.

[8] M.P. Baldado Jr, E.L. Enriquez, Super secure domination in graphs, International Journal of Mathematical Archive, 8 (12) (2017), 145-149.

[9] M.P. Baldado, G.M. Estrada, and E.L. Enriquez, Clique Secure Domination in Graphs Under Some Operations, International Journal of Latest
Engineering Research and Applications, 3(6) (2018), pp 08-14.

[10] E.L. Enriquez, E. Samper-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.

[11] 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, 4(7) (2019), pp 11-16.

[12] E.L. Enriquez, V. Fernandez, Teodora Punzalan, and Jonecis Dayap, Perfect Outer-connected Domination in the Join and Corona of Graphs,
Recoletos Multidisciplinary Research Journal, 4(2) (2016) pp 1-8.

[13] J.A. Telle, A. Proskurowski, Algorithms for Vertex Partitioning Problems on Partial 𝑘 Trees, SIAM J. Discrete Mathematics, 10(1997), 529-550.

[14] Enriquez, E.L., (2020), Fair Restrained Domination in Graphs, International Journal of Mathematics Trends and Technology, 66(1), pp. 229-235.

[15] Galleros, DH.P. and Enriquez, E.L., (2020),Fair Restrained Dominating Set in the Corona of Graphs, International Journal of Engineering and
Management Research, 10(03), pp. 110-114 , https://doi.org/10.31033/ijemr.10.3.17

[16] T.J. Punzalan, and E.L. Enriquez, Restrained Secure Domination in the Join and Corona of Graphs, Journal of Global Research in Mathematical
Archives, 5(5) (2018), pp 01-06.

[17] E.M. Kiunisala, and E.L. Enriquez, Inverse Secure Restrained Domination in the Join and Corona of Graphs, International Journal of Applied
Engineering Research, 11(9) (2016) pp. 6676-6679.

[18] C.M. Loquias, and E.L. Enriquez, On Secure Convex and Restrained Convex Domination in Graphs, International Journal of Applied Engineering
Research, 11(7) (2016) pp. 4707-4710.

[19] E.L. Enriquez, Secure restrained convex domination in graphs, International Journal of Mathematical Archive, 8(7) (2017) pp. 1-5.

[20] B.F. Tubo and S.R. Canoy, Jr., Restrained Perfect Domination in Graphs, International Journal of Mathematical Analysis, 9(25) (2015), pp. 1231 –
1240

[21] T.J. Punzalan, E.L. Enriquez, Inverse Restrained Domination in Graphs, Global Journal of Pure and Applied Mathematics. 12(3) (2016), pp. 2001 -
2009.

[22] V.R. Kulli and S.C. Sigarkanti, Inverse domination in graphs, Nat.Acad. Sci. Letters, 14 (1991) 473-475.

[23] E.L. Enriquez, E.M. Kiunisala, Inverse Secure Domination in the Join and Corona of Graphs, Global Journal of Pure and Applied Mathematics,
12(2) (2016), pp. 1537-1545

[24] E.L. Enriquez, E.M. Kiunisala, Inverse Secure Domination in Graphs, Global Journal of Pure and Applied Mathematics, 12(1) (2016), pp. 147-155

[25] Salve, D.P., and Enriquez, E.L., (2016), Inverse Perfect Domination in the Composition and Cartesian Product of Graphs, Global Journal of Pure
and Applied Mathematics, 12(1), pp. 1-10.

[26] C.M. Loquias, E.L. Enriquez, J.A. Dayap, Inverse Clique Domination in Graphs, Recoletos Multidisciplinary Research Journal, 4(2) (2016) pp 21-
32.
[27] Kiunisala, E.M. and Enriquez, E.L., (2016), Inverse Secure Restrained Domination in the Join and Corona of Graphs, International Journal of
Applied Engineering Research, 11(9), pp. 6676-6679.

[28] D.P. Salve and E.L. Enriquez, Inverse Perfect Domination in Graphs, Global Journal of Pure and Applied Mathematics, 12(1) (2016), pp. 1-10

[29] R.C. Alota and E.L. Enriquez, On Disjoint Restrained Domination in Graphs, Global Journal of Pure and Applied Mathematics, 12(3) (2016), pp.
2385-2394

[30] J.A. Dayap and E.L. Enriquez, Disjoint Secure Domination in the Join of Graphs, Recoletos Multidisciplinary Research Journal, 4(2) (2016) pp 9-
20.
[31] H.R.A. Gohil and E.L. Enriquez, Inverse Perfect Restrained Domination in Graphs, International Journal of Mathematics Trends and Technology,
67(8) (2021) pp 164-170

[32] M.D. Garol and E.L. Enriquez, Disjoint Fair Domination in Graphs, International Journal of Mathematics Trends and Technology, 67(8) (2021) pp
157-163

[33] C.S. Castañares and E.L. Enriquez, Inverse Perfect Secure Domination in Graphs, International Journal of Mathematics Trends and Technology,
67(8) (2021) pp. 150-156

Stephen Paul G. Cajigas, Enrico L. Enriquez, Grace M. Estrada, Katrina E. Belleza, Carmelita M. Loquias, "Disjoint Restrained Domination in the Join and Corona of Graphs," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 67, no. 12, pp. 57-61, 2021. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V67I12P506