Open Packing in H-free Graphs and Subclasses of Split Graphs

Abstract

Given a graph G(V, E), a vertex subset S of G is called an open packing in G if no pair of distinct vertices in S have a common neighbour in G. The size of a largest open packing in G is called the open packing number of G and is denoted by $\rho ^o(G)$. It is interesting to note that the open packing number is a lower bound for the total domination number in graphs with no isolated vertices [Henning and Slater, 1999]. Given a graph G and a positive integer k, Open Packing problem tests whether G has an open packing of size at least k.

It is known that Open Packing is NP-complete on split graphs (i.e., the class of $\{2K_2,C_4,C_5\}$-free graphs) [Ramos et al. 2014]. In this work, we complete the study on the complexity of Open Packing on H-free graphs for every graph H on at least three vertices by proving that Open Packing is (i) NP-complete on $K_{1,3}$-free graphs and (ii) polynomial-time solvable on $(P_4\cup rK_1)$-free graphs for every $r\ge 1$. Further, we prove that Open Packing is (i) NP-complete on $K_{1,4}$-free split graphs and (ii) polynomial-time solvable on $K_{1,3}$-free split graphs. We prove a similar dichotomy result on split graphs with degree restrictions on the vertices in the independent set of the clique-independent set partition of the split graph.