Domination parameters of the splitting graph of a graph

Document Type : Original paper


1 Department of Mathematics, Madurai Kamaraj University, Madurai, Tamilnadu, India

2 Department of Mathematics, The Madura College, Madurai, Tamilnadu, India

3 Birla Institute of Technology and Sciences Pilani, Dubai Campus

4 Director (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626 126 Tamil Nadu, India


Let $G=(V,E)$ be a graph of order $n$ and size $m.$ The graph $Sp(G)$ obtained from $G$ by adding a new vertex $v'$ for every vertex $v\in V$ and joining $v'$ to all neighbors of $v$ in $G$ is called the splitting graph of $G.$ In this paper, we determine the domination number, the total domination number, connected domination number, paired domination number and independent domination number for the splitting graph $Sp(G).$


Main Subjects

[1] M.A. Abdlhusein and S.J. Radhi, The arrow edge domination in graphs, Int. J. Nonlinear Anal. Appl. (In press).
[2] R.B. Allan and R. Laskar, On domination and independent domination numbers of a graph, Discrete Math. 23 (1978), no. 2, 73–76.
[3] L.W. Beineke and J.S. Bagga, Line Graphs and Line Digraphs, Springer, 2021.
[4] G. Chartrand and L. Lesniak, Graphs & Digraphs, CRC, Boca Raton, 2016.
[5] T.W. Haynes, S.T. Hedetniemi, and P.J. Salter, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.
[6] N. Jafari Rad, D.A. Mojdeh, R. Musawi, and E. Nazari, Total domination in cubic Knödel graphs, Commun. Comb. Optim. 6 (2021), no. 2, 221–230.
[7] R. Khoeilar, H. Karami, M. Chellali, S.M. Sheikholeslami, and L. Volkmann, Nordhaus–Gaddum type results for connected and total domination, RAIRO-Operations Research 55 (2021), S853–S862.
[8] S. Kosari, Z. Shao, X. Shi, S.M. Sheikholeslami, M. Chellali, R. Khoeilar, and H. Karami, Cubic graphs have paired-domination number at most four-seventh of their orders, Discrete Math. (to appear).
[9] Y.S. Kwon, J. Lee, and M.Y. Sohn, Domination parameters in Mycielski graphs, Bull. Korean Math. Soc. 58 (2021), no. 4, 829–836.
[10] E. Sampthkumar and H.B. Walikar, On splitting graph of a graph, J. Karnatak Univ. Sci. 25 (1980), no. 13, 13–16.
[11] J. Topp and L. Volkmann, On graphs with equal domination and independent domination numbers, Discrete Math. 96 (1991), no. 1, 75–80.