On $\gamma$-free, $\gamma$-totally-free and $\gamma$-fixed sets in graphs

Document Type : Original paper

Authors

1 Department of Mathematics, S. D. College, Alappuzha-690 104

2 DOS in Mathematics Manasagangotri

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

Abstract

Let $G=(V,E)$ be a connected graph. A subset $S$ of $V$ is called a $\gamma$-free set if there exists a $\gamma$-set $D$ of $G$ such that $S \cap D= \emptyset$. If further the induced subgraph $H=G[V-S]$ is connected, then $S$ is called a  $cc$-$\gamma$-free set of $G$. We use this concept to identify connected induced subgraphs $H$ of a given graph $G$ such that $\gamma(H) \leq \gamma(G)$. We also introduce the concept of $\gamma$-totally-free and $\gamma$-fixed sets and present several basic results on the corresponding parameters. 

Keywords

References

[1] G. Chartrand, L. Lesniak, and P. Zhang, Graphs & digraphs, CRC, Boca Raton, 2016.
[2] T.W. Haynes, S.T. Hedetniemi, and P.J. Salter, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998.
[3] S.M. Hedetniemi, S.T. Hedetniemi, and R. Reynolds, Combinatorial problems on chessboards: II, Domination in Graphs, Advanced Topics (T.W. Haynes, S.T. Hedetniemi, and P.J. Salter, eds.), Marcel Dekker, Inc., New York, 1998, pp. 133–
192.
[4] 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.
[5] E. Sampathkumar and P.S. Neeralagi, Domination and neighbourhood critical, fixed, free and totally free points, Indian J. Statistics (1992), 403–407.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 10 June 2023

Files

Share

How to cite

Statistics