The k-limited packing and k-tuple domination problems in strongly chordal, P4-tidy and split graphs

The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non negative integer k, a subset B of vertices is a k-limited packing if there are at most k elements of B in the closed neighborhood of every vertex. On the other side, a k-tuple domination set in a graph is a subset of vertices D such that every vertex has at least k elements of D in its closed neighborhood. In this work we first reveal a strong relationship between these notions, and obtain from a result due to Liao and Chang (2002), the polynomiality of the k-limited packing problem for strongly chordal graphs.We also prove that, in coincidence with the case of domination, the k-limited packing problem is NP-complete for split graphs. Finally, we prove that both problems are polynomial for the non-perfect class of P4-tidy graphs, including the perfect classes of P4-sparse graphs and cographs.