Independent (k+1)-domination in k-trees

The problem of independent k-domination is defined as follows: A subset S of the set of vertices of a graph G is called independentk-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.