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.

翻译：给定图$G(V,E)$，顶点子集$S$若满足$S$中任意两个不同顶点在$G$中没有公共邻点，则称$S$为$G$的一个开装填。$G$中最大开装填的大小称为$G$的开装填数，记作$\rho^o(G)$。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$G$是否存在大小至少为$k$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OPEN PACKING在分裂图（即$\{2K_2,C_4,C_5\}$-free图）上是NP完全的[Ramos et al., 2014]。本文通过证明OPEN PACKING在(i) $K_{1,3}$-free图上为NP完全，且(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）的完整研究。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$\rho^o(G)$存在以$r$的线性函数为上界的估计。此外，我们证明在$K_{1,3}$-free图上，以解大小为参数的OPEN PACKING问题是W[1]-完全的，且除非P=NP，否则在$K_{1,3}$-free图上MAX-OPEN PACKING问题难以在$n^{(\frac{1}{2}-\delta)}$（任意$\delta>0$）因子内近似。进一步，我们证明OPEN PACKING在(a) $K_{1,4}$-free分裂图上为NP完全，而在(b) $K_{1,3}$-free分裂图上存在多项式时间解法。我们还在分裂图的团-独立集划分中，对独立集顶点施加度数限制，证明了类似的对偶性结果。