• DOI:10.23638/DMTCS-19-1-1
• Corpus ID: 28910264

A characterization of trees with equal 2-domination and 2-independence numbers

@article{Brause2017ACO,
  title={A characterization of trees with equal 2-domination and 2-independence numbers},
  author={Christoph Brause and Michael A. Henning and Marcin Krzywkowski},
  journal={Discret. Math. Theor. Comput. Sci.},
  year={2017},
  volume={19}
}

• Christoph Brause, Michael A. Henning, Marcin Krzywkowski
• Published 21 April 2016
• Mathematics
• Discret. Math. Theor. Comput. Sci.

A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $\gamma_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $\alpha_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.