Edge open packing: further characterizations

2025-08-03

AI Summary

This paper investigates the structural characterization of the edge open packing number ρₑᵒ(G). (Problem) Specifically, it addresses two central questions: (1) establishing necessary and sufficient conditions for ρₑᵒ(G) = t for arbitrary t ≥ 3; and (2) fully characterizing all graphs satisfying ρₑᵒ(G) = m − 3, where m denotes the number of edges. (Method) The analysis leverages edge adjacency relations and integrates combinatorial construction, extremal reasoning, and case-based classification to systematically uncover topological features of graphs attaining specific values of ρₑᵒ(G). (Contribution/Results) The main innovation lies in overcoming prior limitations—where characterizations were restricted to small or extremal values—by developing a precise, general framework for all t ≥ 3, and completing an exhaustive classification for graphs whose edge open packing number is near its theoretical upper bound. This significantly deepens the theoretical foundation and broadens the applicability of edge open packing theory.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Linked Open Data, Knowledge Graphs & KB Completion

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $G=(V, E)$ be a graph where $V(G)$ and $E(G)$ are the vertex and edge sets, respectively. In a graph $G$, two edges $e_1, e_2in E(G)$ are said to have emph{common edge} $e eq e_1, e_2$ if $e$ joins an endpoint of $e_1$ to an endpoint of $e_2$ in $G$. A subset $Dsubseteq E(G)$ is called an emph{edge open packing set} in $G$ if no two edges in $D$ share a common edge in $G$, and the largest size of such a set in $G$ is known as emph{edge open packing number}, represented by $ρ_{e}^o(G)$. In the introductory paper (Chelladurai et al. (2022)), necessary and sufficient conditions for $ρ_{e}^o(G)=1, 2$ were provided, and the graphs $G$ with $ρ_{e}^o(G)in {m-2, m-1, m}$ were characterized, where $m$ is the number of edges of $G$. In this paper, we further characterize the graphs $G$. First, we show necessary and sufficient conditions for $ρ_{e}^o(G)=t$, for any integer $tgeq 3$. Finally, we characterize the graphs with $ρ_{e}^o(G)=m-3$.

Problem

• Characterize graphs with specific edge open packing numbers
• Establish conditions for edge open packing number equals t
• Identify graphs where edge open packing number is m-3

Innovation

• Defines edge open packing set criteria
• Determines conditions for packing number t
• Characterizes graphs with specific packing numbers