International Journal of Nonlinear Analysis and Applications
Stability of (1,2)-total pitchfork domination
Document Type : Research Paper
Authors
Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq
Abstract
Let $G=(V, E)$ be a finite, simple, and undirected graph without isolated vertex. We define a dominating $D$ of $V(G)$ as a total pitchfork dominating set, if $1\leq|N(t)\cap V-D|\leq2$ for every $t \in D$ such that $G[D]$ has no isolated vertex. In this paper, the effects of adding or removing an edge and removing a vertex from a graph are studied on the order of minimum total pitchfork dominating set $\gamma_{pf}^{t} (G)$ and the order of minimum inverse total pitchfork dominating set $\gamma_{pf}^{-t} (G)$. Where $\gamma_{pf}^{t} (G)$ is proved here to be increasing by adding an edge and decreasing by removing an edge, which are impossible cases in the ordinary total domination number.
Keywords
[2] M. A. Abdlhusein, Applying the (1,2)-pitchfork domination and its inverse on some special graphs, Bol. Soc. Paran. Mat., accepted to appear (2021).
[3] M. A. Abdlhusein, Stability of inverse pitchfork domination, Int. J. Nonlinear Anal. Appl. 12(1) (2021) 1009–1016.
[4] M. A. Abdlhusein and M. N. Al-Harere, Pitchfork domination and it’s inverse for corona and join operations in graphs, Proc. Int. Math. Sci. I(2) (2019) 51–55.
[5] M. A. Abdlhusein and M. N. Al-Harere, New parameter of inverse domination in graphs, Indian J. Pure Appl. Math.(accepted to appear)(2021).
[6] M. A. Abdlhusein and M. N. Al-Harere, Doubly connected pitchfork domination and its inverse in graphs, TWMS J. App. Eng. Math., (accepted to appear) (2021).
[7] M. A. Abdlhusein and M. N. Al-Harere, Some modified types of pitchfork domination and its inverse, Bol. Soc. Paran. Mat., accepted to appear (2021).
[8] M. A. Abdlhusein and M. N. Al-Harere, Total pitchfork domination and its inverse in graphs, Discrete Math. Algo. Appl.(2020) 2150038.
[9] M. A. Abdlhusein and M. N. Al-Harere, Pitchfork domination and its inverse for complement graphs, Proc. IAM, 9(1) (2020) 13–17.
[10] M. N. Al-Harere and M. A. Abdlhusein, Pitchfork domination in graphs, Discrete Math. Algo. Appl. 12(2) (2020) 2050025.
[11] M. N. Al-Harere and A. T. Breesam, Variant types of domination in spinner graph, Al-Nahrain J. 2 (2019) 127–133.
[12] M. N. Al-Harere and P. A. Khuda Bakhash, Changes of tadpole domination number upon changing of graphs, Sci. Int. 31(2) (2019) 197–199.
[13] M. N. Al-Harere and P. A. Khuda Bakhash, Tadpole domination in duplicated graphs, Discrete Math. Algo. Appl. 13(2) (2021) 2150003.
[14] M. Amraee, N. J. Rad and M. Maghasedi, Roman domination stability in graphs, Math. Rep. 21(71) (2019) 193–204.
[15] B. A. Atakul, Stability and domination exponentially in some graphs, AIMS Math. 5(5)(2020) 5063–5075.
[16] K. Attalah and M. Chellali, 2-Domination dot-stable and dot-critical graphs, Asian-European J. Math. 21(5) (2021) 2150010.
[17] S. Balamurugan, Changing and unchanging isolate domination: edge removal, Discrete Math. Algo. Appl. 9(1) (2017) 1750003.
[18] A. Das, R. C. Laskar and N. J. Rad, On α-domination in graphs, Graph. Comb. 34(1) (2018) 193–205.
[19] W. J. Desormeaux, T. W. Haynes and M. A. Henning, Total domination critical and stable graphs upon edge removal, Discrete Appl. Math. 158 (2010) 1587–1592.
[21] F. Harary, Graph Theory, Addison-Wesley, Reading Mass, 1969.
[22] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker Inc., New York, (1998).
[23] T. W. Haynes, S. T. Hedetniemi and P.J. Slater, Domination in Graphs-Advanced Topics, Marcel Dekker Inc., 1998.
[24] S. T. Hedetneimi and R. Laskar (Eds.), Bibliography on domination in graphs and some basic definitions of domination parameters, Discrete Math. 86 (1990) 257–277.
[25] M. A. Henning and M. Krzywkowski, Total domination stability in graphs, Discrete Appl. Math. 236(19)(2018) 246–255.
[26] A. A. Omran and T. A. Ibrahim, Fuzzy co-even domination of strong fuzzy graphs, Int. J. Nonlinear Anal. Appl. 12(1) (2021) 727–734.
[27] O. Ore, Theory of graphs, American Mathematical Society, Provedence, R.I., (1962).
[28] S. J. Radhi, M. A. Abdlhusein and A. E. Hashoosh, The arrow domination in graphs, Int. J. Nonlinear Anal. Appl. 12(1) (2021) 473–480.
[29] V. Samodivkin, A note on Roman domination: changing and unchanging, Aust. J. Comb. 71(2) (2018) 303–311.
)
Volume 12, Issue 2
November 2021Pages 265-274
- Receive Date: 25 December 2020
- Revise Date: 18 February 2021
- Accept Date: 03 March 2021