### Abstract

The problem of independent k-domination is defined as follows: A subset S of the set of vertices of a graph G is called independent k-dominating in G, if S is both independent and k-dominating. In 2003, Haynes, Hedetniemi, Henning and Slater studied this problem in the class of trees, and gave the characterization of all trees having an independent 2-dominating set. They also proved that if such a set exists, then it is unique. We extend these results to k-degenerate graphs and k-trees as follows. We prove that if a k-degenerate graph has an independent (k+1)-dominating set, then this set is unique; moreover, we provide an algorithm that tests whether a k-degenerate graph has an independent (k+1)-dominating set and constructs this set if it exists. Next we focus on independent 3-domination in 2-trees and we give a constructive characterization of 2-trees having an independent 3-dominating set. Using this, tight upper and lower bounds on the number of vertices in an independent 3-dominating set in a 2-tree are obtained.

Original language | English |
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Journal | Discrete Applied Mathematics |

DOIs | |

Publication status | Accepted/In press - Jan 1 2020 |

### Keywords

- 2-tree
- Independent 3-domination
- Independent domination
- k-domination

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2020.03.019