Is the minimum weight independent dominating set np-complete in chordal graphs? I have a found a small article [1] saying (the first paragraph of the introduction) that the minimum-weight independent dominating set is NP-complete in chordal graphs, but at the same time, seems to contradict that exact statement.

Moreover, I have found another reference [2] saying that in chordal graphs, it is polynomial time solvable. So which one is it?

Note: I am just trying to reference this result in a project of mine. No need for a proof.

Edit: I am referring to this piece of the introduction: "Domination and most of its variations are NP-complete for chordal graphs (even for the subclass of split graphs) with the exception of independence domination (see [3]). On the other hand, an unpublished proof for the NP-completeness of the weighted independent domination in chordal graphs by the author 20 years ago..." Then in my reference [2], it also states that the weighted version is polynomial time solvable, yet here they say that there is an NP-completeness proof. Am I missing something fundamental?




The weighted independent domination problem is NP-complete for chordal graphs by G. J. Chang (2004)
Fundamentals of Domination in Graphs by T. W. Haynes, S. Hedetniemi and P. Slater (1998)
As the article clearly mentions, unweighted minimum independent dominating set can be solved efficiently (in fact, linear time) for chordal graphs. A glance at the paper shows that this works even if the vertices are given $\{0,1\}$ weights (rather than all having unit weight). On the other hand, weighted minimum independent dominating set is NP-complete.