Edge open packing: Complexity, algorithmic aspects, and bounds

Abstract

Given a graph G, two edges e1,e2∈E(G) are said to have a common edge e≠e1,e2 if e joins an endvertex of e1 to an endvertex of e2. A subset B⊆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, ρeo(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 (Chelladurai et al. (2022) [5]). Notably, we characterize the graphs G that attain the upper bound ρeo(G)≤|E(G)|/δ(G), and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.