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.

中文翻译：

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k 树中的独立 (k+1)-支配

摘要 独立k-支配问题定义如下：如果S既独立又k-支配，则图G的顶点集的子集S称为G中的独立k-支配。2003 年，Haynes、Hedetniemi、Henning 和 Slater 在树类中研究了这个问题，并给出了所有具有独立 2 支配集的树的表征。他们还证明了如果这样一个集合存在，那么它就是唯一的。我们将这些结果扩展到 k-退化图和 k-树，如下所示。我们证明如果 k -退化图有一个独立的 ( k + 1 ) -支配集，那么这个集是唯一的；此外，我们提供了一种算法来测试 k -退化图是否具有独立的 ( k + 1 ) -支配集，如果存在则构造该集。接下来，我们关注 2 棵树中的独立 3 支配，我们给出了具有独立 3 支配集的 2 树的建设性特征。使用此方法，可以获得 2 树中独立 3 支配集中的顶点数量的严格上限和下限。