Abstract. In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the k-tuple domination problem in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the 2-tuple domination problem in trees by employing a labeling method. 1.

The concept of domination in graph theory is a natural model for many location problems in operations research. In a graph G = (V, E), a dominating set is a subset D of V such that every vertex not in D is adjacent to at least one vertex in D. The domination problem is to determine, for a graph G and a positive integer k, if G has a dominating set of size at most k. Domination and its variations have been extensively studied in the literature (see [1, 4, 5]). 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 On the other hand, an unpublished proof for the NP-completeness of the weighted independent domination in chordal graphs by the author twenty years ago (see [3, 2]) has been queried from time to time. Someone even claimed that he has an efficient algorithm for the problem. The purpose of this paper is to make a record of the proof for those who are interested in. Recall that an independent dominating set is a dominating set whose elements are pairwise non-adjacent. For the weighted version, each vertex is associated with a