Edge open packing on subclasses of chordal graphs

Packing problems on graphs are fundamental in combinatorial optimization and arise naturally in applications such as resource allocation, scheduling, and communication networks. A classical example is the \emph{induced matching} problem, which asks for a set of edges $M\subseteq E(G)$ such that the subgraph induced by the endpoints of $M$ forms a matching. In 2022, Chelladurai et al. (RAIRO-Operations Research, 2022.) introduced the notion of \emph{edge open packing}, which generalizes induced matchings by requiring that no two chosen edges have a common edge joining their endpoints.      For a graph $G=(V,E)$, two edges $e_1,e_2\in E(G)$ are said to have a common edge if there exists an edge $e\in E(G)\setminus\{e_1,e_2\}$ joining an endpoint of $e_1$ to an endpoint of $e_2$. A set $D\subseteq E(G)$ is an \emph{edge open packing} if no two edges in $D$ have a common edge, and the maximum cardinality of such a set is the \emph{edge open packing number} $\rho_e^o(G)$. The corresponding optimization problem is the \textsc{Maximum Edge Open Packing Problem}.       In this paper, we study the computational complexity of the \textsc{Maximum Edge Open Packing Problem}. Motivated by an open question posed by Bre{\v{s}}ar and Samadi (Theor. Comput. Sci., 2024) regarding chordal graphs, we investigate the problem on several subclasses of chordal graphs. We show that the problem can be solved in polynomial time on \emph{proper interval graphs} and \emph{block graphs}. Moreover, we present a linear-time algorithm for the problem on \emph{split graphs}. These results provide partial answers to the open question and contribute to the algorithmic understanding of edge packing parameters in chordal graph classes.