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Given a positive integer k, the "{k}-packing function problem" ({k}PF) is to find in a given graph G, a function f that assigns a nonnegative integer to the vertices of G in such a way that the sum of f (v) over each closed neighborhood is at most k and over the whole vertex set of G (weight of f ) is maximum. It is known that {k}PF is linear time solvable in strongly chordal graphs and in graphs with clique-width bounded by a constant. In this paper we prove that {k}PF is NP-complete, even when restricted to chordal graphs that constitute a superclass of strongly chordal graphs. To find other subclasses of chordal graphs where {k}PF is tractable, we prove that it is linear time solvable for doubly chordal graphs, by proving that it is so in the superclass of dually chordal graphs, which are graphs that have a maximum neighborhood ordering.

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-dominating set, in a given graph, are referred to as k-domination, resp. total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from recent work by Kang et al. (2017) that these two families of problems are solvable in time O(|V (G)| 6k+4 ) in the class of interval graphs. In this work, we develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs. The algorithms run in time O(|V (G)| 3k ) for each fixed k ≥ 1 and are also applicable to the weighted case.