Open Packing in Graphs: Bounds and Complexity

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, $\rho^o(G)$, of $G$. It would be 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 $k$, the decision problem OPEN PACKING tests whether $G$ has an open packing of size at least $k$. The optimization problem MAX-OPEN PACKING takes a graph $G$ as input and finds the open packing number of $G$. It is known that OPEN PACKING is NP-complete on split graphs (i.e., $\{2K_2,C_4,C_5\}$-free graphs) [Ramos et al., 2014]. In this work, we complete the study on the complexity (P vs NPC) of OPEN PACKING on $H$-free graphs for every graph $H$ with 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 $r\geq 1$. In the course of proving (ii), we show that for every $t\in {2,3,4}$ and $r\geq 1$, if G is a $(P_t\cup rK_1)$-free graph, then $\rho^o(G)$ is bounded above by a linear function of $r$. Moreover, we show that OPEN PACKING parameterized by solution size is W[1]-complete on $K_{1,3}$-free graphs and MAX-OPEN PACKING is hard to approximate within a factor of $n^{(\frac{1}{2}-\delta)}$ for any $\delta>0$ on $K_{1,3}$-free graphs unless P=NP. Further, we prove that OPEN PACKING is (a) NP-complete on $K_{1,4}$-free split graphs and (b) 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 graphs.