Open Packing in Interval Graphs

Abstract

Total Domination and Open Packing forms a primal-dual pair of problems. A vertex subset S of a graph G is called an open packing in G if no pair of distinct vertices in S have a common neighbour in G. The cardinality of a maximum open packing in G is called the open packing number, ρᵒ(G), of G which is a lower bound for the total domination number of G. Given a graph G and a positive integer k, the problem OPEN PACKING tests whether G has an open packing of size at least k. It is known that OPEN PACKING is NP-complete for split graphs (and so for chordal graphs) [Ramos et al., 2014]. In this work, we use a dynamic programming based approach to show that a maximum open packing in interval graphs (a subclass of chordal graphs) can be found in O(n³) time.