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 a \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 the \emph{edge open packing number}, represented by $ρ_{e}^o(G)$. The \textsc{Maximum Edge Open Packing Problem} is to find an edge open packing set of a given graph with maximum size. In [Bre{š}ar and Samadi. Edge open packing: complexity, algorithmic aspects, and bounds. Theor. Comput. Sci., 2024.], Bre{š}ar and Samadi pose an open question of the edge open packing problem in chordal graphs. In this paper, we partially answer this open question by showing a polynomial-time algorithm to solve the maximum edge open packing problem in the subclasses of chordal graphs. First, we show that the \textsc{Maximum Edge Open Packing Problem} can be solved in polynomial time for \emph{proper interval graphs}. Furthermore, we show that in \emph{block graphs} we can solve this problem in polynomial time. Finally, we prove that this problem can be solved in linear time for \emph{split graphs}.

翻译：设 $G=(V, E)$ 为一个图，其中 $V(G)$ 和 $E(G)$ 分别为其顶点集和边集。在图 $G$ 中，若存在一条边 $e\neq e_1, e_2$ 将 $e_1$ 的一个端点与 $e_2$ 的一个端点相连，则称两条边 $e_1, e_2\in E(G)$ 在 $G$ 中具有一条\emph{公共边}。若 $G$ 中不存在任意两条边共享一条公共边，则子集 $D\subseteq E(G)$ 称为 $G$ 中的一个\emph{边开包装集}，此类集合在 $G$ 中的最大规模称为\emph{边开包装数}，记为 $ρ_{e}^o(G)$。\textsc{最大边开包装问题}旨在寻找给定图中具有最大规模的边开包装集。在 [Bre{š}ar 和 Samadi. Edge open packing: complexity, algorithmic aspects, and bounds. Theor. Comput. Sci., 2024.] 中，Bre{š}ar 和 Samadi 提出了关于弦图中边开包装问题的一个开放性问题。本文通过展示在弦图子类中求解最大边开包装问题的多项式时间算法，部分回答了该开放性问题。首先，我们证明对于\emph{真区间图}，\textsc{最大边开包装问题}可在多项式时间内求解。此外，我们证明在\emph{块图}中该问题同样可在多项式时间内求解。最后，我们证明对于\emph{分裂图}，该问题可在线性时间内求解。