Given a graph $G$, two edges $e_{1},e_{2}\in E(G)$ are said to have a common edge $e$ if $e$ joins an endvertex of $e_{1}$ to an endvertex of $e_{2}$. A subset $B\subseteq E(G)$ is an edge open packing set in $G$ if no two edges of $B$ have a common edge in $G$, and the maximum cardinality of such a set in $G$ is called the edge open packing number, $\rho_{e}^{o}(G)$, of $G$. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree $4$, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper [Edge open packing sets in graphs, RAIRO-Oper.\ Res.\ 56 (2022) 3765--3776]. Notably, we characterize the graphs $G$ that attain the upper bound $\rho_e^o(G)\le |E(G)|/\delta(G)$, and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.

翻译：给定图$G$，若边$e$连接$e_{1}$的一个端点与$e_{2}$的一个端点，则称边$e_{1},e_{2}\in E(G)$有公共边$e$。子集$B\subseteq E(G)$称为$G$中的边开放打包集，若$B$中任意两条边在$G$中没有公共边，$G$中此类集合的最大基数称为边开放打包数，记为$\rho_{e}^{o}(G)$。本文证明：即使分别限制在具有泛顶点的图、欧拉二部图和最大度为$4$的平面图，边开放打包数的判定版本仍是NP完全的。与此相对，我们提出了一个线性时间算法来计算树的边开放打包数。同时解决了开创性论文[Edge open packing sets in graphs, RAIRO-Oper.\ Res.\ 56 (2022) 3765--3776]中提出的两个问题。特别地，我们刻画了达到上界$\rho_e^o(G)\le |E(G)|/\delta(G)$的图$G$，给出了图的边删除子图的下界与上界，并建立了相应的可实现性结果。