Abstract
For a graph $G$, denote its domination number and independent domination number by $\gamma(G)$ and $i(G)$. For a positive integer $k$, a subset $S \subseteq V(G)$ is a $k$-independent set of $G$ if $\Delta(G[S]) \lt k$. The $k$-independence number $\alpha_{k}(G)$ is the cardinality of a maximum $k$-independent set of $G$. In this paper, we prove that for any nontrivial tree $T$, $i(T) \leq \frac{k+1}{2k} \alpha_{k}(T)$ and $\gamma(T) \leq \frac{k}{2k-1} \alpha_{k}(T)$, respectively. This extends the recent result for $k = 2$ by Dehgardi et al. Both bounds are best possible and the extremal trees are characterized.
Funding Statement
This work is supported by XJEDU2019I001 and NSFC (Nos. 12061073, 11801487).
Acknowledgments
We would like to thank the two anonymous reviewers and the editor for their careful reading and helpful comments.
Citation
Gang Zhang. Baoyindureng Wu. "Domination, Independent Domination and $k$-independence in Trees." Taiwanese J. Math. 26 (2) 221 - 231, April, 2022. https://doi.org/10.11650/tjm/211005
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