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 an 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

[1] M.A. Abdlhusein, Doubly connected bi-domination in graphs, Discrete Math. Algorithms Appl. 13 (2021), no. 2, 2150009.
[2] M.A. Abdlhusein, Stability of inverse pitchfork domination, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 1009–1016.
[3] M.A. Abdlhusein, Applying the (1,2)-pitchfork domination and its inverse on some special graphs, Bol. Soc. Paran. Mat. 41 (2023), 1–8.
[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. 1 (2019), no. 2, 51–55.
[5] M.A. Abdlhusein and M.N. Al-Harere, Pitchfork domination and its inverse for complement graphs, Proc. IAM 9 (2020), no. 1, 13–17.
[6] M.N. Al-Harere and M.A. Abdlhusein, Pitchfork domination in graphs, Discrete Math. Algorithms Appl. 12 (2020), no. 2, 2050025.
[7] M.A. Abdlhusein and M.N. Al-Harere, New parameter of inverse domination in graphs, Indian J. Pure Appl. Math. 52 (2021), no. 1, 281–288.
[8] M.A. Abdlhusein and M.N. Al-Harere, Total pitchfork domination and its inverse in graphs, Discrete Math. Algorithms Appl. 13 (2021), no. 4, 2150038.
[9] M.A. Abdlhusein and M.N. Al-Harere, Doubly connected pitchfork domination and its inverse in graphs, TWMS J. App. Eng. Math. 12 (2022), no. 1, 82.
[10] M.A. Abdlhusein and M.N. Al-Harere, Some modified types of pitchfork domination and its inverse, Bol. Soc. Paran. Mat. 40 (2022), 1–9.
[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 (2019), no. 2, 197–199.
[13] M.N. Al-Harere and P.A. Khuda Bakhash, Tadpole domination in duplicated graphs, Discrete Math. Algorithms Appl. 13 (2021), no. 2, 2150003.
[14] M. Amraee, N.J. Rad, and M. Maghasedi, Roman domination stability in graphs, Math. Rep. 21 (2019), no. 71, 193–204.
[15] B.A. Atakul, Stability and domination exponentially in some graphs, AIMS Math. 5 (2020), no. 5, 5063–5075.
[16] K. Attalah and M. Chellali, 2-Domination dot-stable and dot-critical graphs, Asian-Eur. J. Math. 21 (2021), no. 5, 2150010.
[17] S. Balamurugan, Changing and unchanging isolate domination: edge removal, Discrete Math. Algorithms Appl. 9 (2017), no. 1, 1750003.
[18] A. Das, R.C. Laskar, and N.J. Rad, On α-domination in graphs, Graphs Combin. 34 (2018), no. 1, 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.
[20] W.J. Desormeaux, T.W. Haynes, and M.A. Henning, Total domination stable graphs upon edge addition, Discrete Math. 310 (2010), 3446–3454.
[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. Hedetniemi and R. Laskar, Topics in domination in graphs, Discrete Math. 86 (1990), no. 1-3, 3–9.
[25] M.A. Henning and M. Krzywkowski, Total domination stability in graphs, Discrete Appl. Math. 236 (2018), no. 19, 246–255.
[26] A.A. Omran and T.A. Ibrahim, Fuzzy co-even domination of strong fuzzy graphs, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 727–734.
[27] O. Ore, Theory of Graphs, American Mathematical Society, Providence, 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 (2021), no. 1, 473–480.
[29] V. Samodivkin, A note on Roman domination: Changing and unchanging, Austr. J. Combin. 71 (2018), no. 2, 303–11.
Volume 16, Issue 4
April 2025
Pages 233-240
  • Receive Date: 24 February 2021
  • Accept Date: 25 April 2021