Pitchfork Domination and It's Inverse for Corona and Join Operations in Graphs

Yıl 2019, Cilt: 1 Sayı: 2, 51 - 55, 29.12.2019

Mohammed A. Abdlhusein Manal N. Al-harere

Öz

Let $G$ be a finite simple and undirected graph without isolated vertices. A subset $D$ of $V$ is a pitchfork dominating set if every vertex $v \in D$  dominates at least $j$ and at most $k$ vertices of $V-D$, where $j$ and $k$  are non-negative integers .The domination number of $G$, denoted by $\gamma_{pf}(G)$ is a minimum cardinality over all pitchfork dominating sets in $G$. A subset $D^{-1}$ of $V-D$ is an  inverse pitchfork dominating set if $D^{-1}$ is a pitchfork dominating set.  The inverse domination number of $G$, denoted by $\gamma_{pf}^{-1}(G)$ is a minimum cardinality over all inverse pitchfork  dominating sets in $G$. In this paper, the pitchfork domination and the inverse pitchfork domination are determined when $j=1$ and $k=2$ for some graphs that obtained from graph operations  corona and join.

Anahtar Kelimeler

pitchfork domination

Kaynakça

• [1] M. A. Abdlhusein and M. N. Al-harere, Pitchfork Domination in Graphs, (2019) reprint.
• [2] M. A. Abdlhusein and M. N. Al-harere, Inverse Pitchfork Domination in Graphs, (2019) reprint.
• [3] M. N. Al-harere and A. T. Breesam, Further Results on Bi-domination in Graphs, AIP Conference Proceedings 2096 (2019) 020013-020013-9p.
• [4] M. N. Al-harere and P. A. Khuda Bakhash, Tadpole Domination in Graphs, Baghdad Science Journal 15 (2018) 466-471.
• [5] M. Chellali, T. W. Haynes, S. T. Hedetniemi, and A. M. Rae, [1,2]-Set in graphs, Discrete Applied Mathematic, 161 (2013) 2885- 2893.
• [6] F. Harary, Graph Theory, Addison-Wesley, Reading Mass, (1969).
• [7] T. W. Haynes, S. T. Hedetniemi and P.J. Slater, Domination in Graphs -Advanced Topics, Marcel Dekker Inc., (1998).
• [8] S.T. Hedetneimi and R. Laskar, Topics in domination in graphs, Discrete Math. 86 (1990).
• [9] A. A. Omran and Y. Rajihy, Some Properties of Frame Domination in Graphs, Journal of Engineering and Applied Sciences, 12 (2017) 8882-8885.
• [10] Ore O. Theory of Graphs, American Mathematical Society, Provedence, R.I. (1962).