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 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., 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 (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.

翻译：给定图$G(V,E)$，顶点子集S称为G中的一个开装填，若S中任意两个不同顶点在G中没有公共邻点。G中最大开装填的大小称为G的开装填数，记为$\rho^o(G)$。值得注意的是，对于无孤立点的图，开装填数是全控制数的一个下界[Henning and Slater, 1999]。给定图G和正整数k，判定问题OPEN PACKING检验G是否具有大小至少为k的开装填。优化问题MAX-OPEN PACKING以图G为输入，求G的开装填数。已知在分裂图（即$\{2K_2,C_4,C_5\}$-free图类）上OPEN PACKING是NP完全的[Ramos et al., 2014]。本工作中，我们通过证明OPEN PACKING在(i) $K_{1,3}$-free图上为NP完全，及(ii) 对所有$r\geq 1$在$(P_4\cup rK_1)$-free图上存在多项式时间解法，完成了对至少含三个顶点的任意图H，OPEN PACKING在H-free图上复杂性（P与NPC）的研究。在证明(ii)的过程中，我们证明了对所有$t\in {2,3,4}$和$r\geq 1$，若G是$(P_t\cup rK_1)$-free图，则$\rho^o(G)$有以r的线性函数表示的上界。此外，我们证明在$K_{1,3}$-free图上，以解大小为参数的OPEN PACKING是W[1]-完全的，且除非P=NP，在$K_{1,3}$-free图上MAX-OPEN PACKING难以在$n^{(\frac{1}{2}-\delta)}$因子内近似（对任意$\delta>0$）。进一步，我们证明OPEN PACKING在(a) $K_{1,4}$-free分裂图上为NP完全，在(b) $K_{1,3}$-free分裂图上存在多项式时间解法。我们在分裂图上关于独立集中顶点度限制的团-独立集划分证明了类似的二分性结果。