Edge open packing: further characterizations

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 $e_1, e_2\in E(G)$ are said to have \emph{common edge} $e\neq e_1, e_2$ if $e$ joins an endpoint of $e_1$ to an endpoint of $e_2$ in $G$. A subset $D\subseteq E(G)$ 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 $\rho_{e}^o(G)$. In the introductory paper (Chelladurai et al. (2022)), necessary and sufficient conditions for $\rho_{e}^o(G)=1, 2$ were provided, and the graphs $G$ with $\rho_{e}^o(G)\in \{m-2, m-1, m\}$ 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 $\rho_{e}^o(G)=t$, for any integer $t\geq 3$. Finally, we characterize the graphs with $\rho_{e}^o(G)=m-3$.