On the trees with same signed edge and signed star domination numbers

ABSTRACT

Let ${\gamma }_S^{\prime }\left(G\right)$ and ${\gamma }_{SS}\left(G\right)$ be the signed edge domination number and signed star domination number of a graph G, respectively. In this paper, we prove that for a tree T of order $n\ge 3$, if ${\gamma }_S^{\prime }\left(T\right)={\gamma }_{SS}\left(T\right)=k$ then $k\ge (3n+1)/5$. Moreover, we prove that for any integer $r\ge 1$, there exists a tree $T_r$ on n=5r+3 vertices such that ${\gamma }_S^{\prime }\left(T_r\right)={\gamma }_{SS}\left(T_r\right)=(3n+1)/5$.

KEYWORDS:

• Tree
• signed edge domination function
• signed edge domination number
• signed star domination function
• signed star domination number

2010 AMS SUBJECT CLASSIFICATION:

• 05C05
• 05C69

Acknowledgments

The authors would like to thank the referees for valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Key Research and Development Program of China [grant number 2016YFB0800700] and National Natural Science Foundation of China [grant numbers 61672050, 61632002, 61502012 and 61572046].