Edge open packing: complexity, algorithmic aspects, and bounds

TL;DR

The paper studies the edge open packing problem, introducing the EOP number $\rho_e^{o}(G)$ that captures the maximum size of an edge set with no common edge between any two members. It proves NP-completeness of the decision version in restricted graph classes, presents a linear-time dynamic-programming algorithm for computing $\rho_e^{o}(T)$ on trees, and establishes extremal and realization results, including a complete characterization of graphs achieving the bound $\rho_e^{o}(G)\le |E(G)|/\delta(G)$. It also analyzes how edge deletion affects $\rho_e^{o}$, delivering tight bounds and showing all intermediate values can be realized. The work connects edge open packing with injective edge coloring and provides practical algorithmic tools for trees while clarifying the combinatorial structure of general graphs.

Abstract

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, $ρ_{e}^{o}(G)$, 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 $ρ_e^o(G)\le |E(G)|/δ(G)$, and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.

Figures (1)

• Figure 1: The subgraph of $H_{G}$ corresponding to the adjacent vertices $v_{i}$ and $v_{j}$ of $G$.

Theorems & Definitions (20)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof