On the complexity of the { k }‐packing function problem

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.