Edge open packing: Complexity, algorithmic aspects, and bounds

Given a graph G, two edges $e_1,e_2\in E\left(G\right)$ are said to have a common edge $e\ne e_1,e_2$ if e joins an endvertex of $e_1$ to an endvertex of $e_2$. A subset $B\subseteq E\left(G\right)$ 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, ${\rho }_e^o\left(G\right)$, 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 (Chelladurai et al. (2022) [5]). Notably, we characterize the graphs G that attain the upper bound ${\rho }_e^o\left(G\right)\le \left|E\right(G\left)\right|/\delta \left(G\right)$, and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.