Inverse planar domination and independent planar domination in graphs
Abstract
Suppose that \(G\) is a simple, undirected, finite graph with no isolated vertices. A dominating set \(D\) of \(G\) is said to be planar dominating set if \(\left\langle D \right\rangle\ \)is a planar graph, such that all vertices in \(D\) have at least two adjacent vertices in \(G\). The planar domination number of \(G\) denoted by \(\gamma_{pl}\), is the cardinality of the minimum planar dominating set in \(G\). Inverse and independent planar domination, the new models of domination are studied in this paper with several results and properties. The inverse planar domination number is evaluated and proved for some known graphs. Also, some possible applications of the planar domination are presented.
Downloads
References
D.A. Abbass, M. N. Al-Harere and E. B. Al-Zangana, On planar domination in graphs, reprint, (2025).
M. N. Al-Harere and A.T. Breesam, Further results on bi-domination in graphs, AIP Conf. Proc.,1 ,020013, (2019).
M. N. Al-Harere, R.J. Mitlif and F.A.Sadiq, Variant domination types for a complete h-ary tree, Baghdad Science Journal, 18m 797-802,(2021).
M. A. A. Abdlhusein and M. N. Al-Harere, Doubly connected pitchfork domination and its inverse in graphs, Turkish World Mathematical Society Journal of Applied and Engineering Mathematics,12, 82-91, (2022).
M. N. Al-Harere and P. A. Bakhash Khuda, Tadpole domination in duplicated graphs, Discrete Mathematics, Algorithms and Applications, 13, 2150003, (2021).
M. N. Al-Harere and P.A. Khuda Bakhash,Changing and unchanging on tadpole domination in G − e, G + e graphs, Turkish World Mathematical Society Journal of Applied and Engineering Mathematics,13,1151-1159, (2022).
K.S. Al’Dzhabri, A.A. Omran and M.N. Al-Harere, DG-domination topology in digraph, Journal of Prime Research in Mathematics, 17, 93-100, (2021).
M.A. Abdlhusein and M.N. Al-Harere, Some modified types of pitchfork domination and its inverse, Boletim da Sociedade Paranaense de Matematica, 40, 1–9, (2022).
E.B. Al-Zangana. and E.A.S. Shehab, Conic Parameterization in PG(2,25), Al-Mustansiriyah Journal of Science, 29,164-168, (2018).
M.N. Al-Harere and P.A Khuda Bakhash,Tadpole domination in duplicated graphs,Discrete Mathematics, Algorithms and Applications 13, 2150003,(2021).
S. M. Attook and E. B. Al-Zangana, Special Arcs in PG(3,13), Al-Mustansiriyah Journal of Science, 33, 286-91, (2022).
H. H Aljanaby and A. A. Omran, Fuzzy hn-domination of strong fuzzy graphs, AIP Conference Proceedings ,2386, 060001 (2022).
C. Berge, The theory of graphs and its applications, Methuen, (1962).
F. Harary ,Graph theory. London: Addison-Wesley, (1969).
T. W. Haynes, S. T. Hedetniemi and P.J. Slater, Domination in Graphs Advanced Topics, Marcel Dekker Inc., (1998).
S.T. Hedetneimi and R. Laskar, Topics in Domination in Graphs, Discrete Math., (1990).
S. SH. Kahat and M. N.Al-Harere,Total equality Co-neighborhood domination in a graph, AIP Conference Proceedings,2414 ,1,(2023).
S. SH. Kahat and M. N. Al-Harere,Dominating sets and polynomial of equality co-neighborhood domination of graphs, AIP Conference Proceedings,1,2398, (2022).
S.S. Kahat and M.N. Al-Harere,Inverse equality co-neighborhood domination in graphs, Journal of Physics: Conference Series 1879, 3, 032036, (2021).
S. S. Kahat, A. A. Omran and M. N. Al-Harere, Fuzzy equality co-neighborhood domination of graphs, Int. J. Nonlinear Anal. Appl., 12, 537-545, (2021).
A. A. Jabor and A. A. Omran, Topological domination in graph theory, In AIP Conference Proceedings, 2334, 020010, (2021).
S.S. Majeed, A.A. Omran and M.N. Yaqoob,Modern Roman domination of corona of cycle graph with some certain graphs, International Journal of Mathematics and Computer Science,17, 317–324, (2022).
S. Salah, A. A. Omran and M. N. Al-Harere,Modern roman domination on two operations in certain graphs, AIP Conf. Proc., 2386,(2022).
M.M. Shalaan and A.A. Omran, Co-Even Domination number in some graphs, IOP Conf. Ser.: Mater. Sci. Eng., 928, 042015, (2020) .
A. A. Omran and T. A. Ibrahim, Fuzzy co-even domination of strong fuzzy graphs, International Journal of Nonlinear Analysis and Applications, 12, 727-734, (2021).
A. A.Omran, M.N.Al-Harere, and S.S.Kahat, Equality co-neighborhood domination in graphs, Discrete Mathematics, Algorithms and Applications, 14,2150098, (2022).
O. Ore, Theory of graphs, American Mathematical Soc. 1962.
J. S. Radhi and E. B. Al-Zangana,Construction of Complete Kr− Arcs from Orbits in PG (3,11) , Al-Mustansiriyah Journal of Science, 33, 48-53, (2022).
S. S. Abed and Manal Naji Al-Harere,On rings domination in graphs, International Journal of Mathematics and Computer Science,17,1313–1319, (2022).
S. S. Abed and Manal Naji Al-Harere,Rings domination in graphs, International Journal of Nonlinear Analysis and Applications,13,1833–1839, (2022).
H. J. Yousif and A. A. Omran ,2-anti fuzzy domination in anti-fuzzy graphs, IOP Conf. Ser.: Mater. Sci. Eng., 928, 042027, (2020).
H.J. Yousif and A.A. Omran, Closed fuzzy dominating set in fuzzy graphs, Journal of Physics: Conference Series, 3, 032022,(2021).
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
