Edge open packing on subclasses of chordal graphs

Packing problems in 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, where one seeks a set of edges whose endpoints induce a matching. In 2022, Chelladurai et al. introduced the notion of \emph{edge open packing}, which can be viewed as a relaxation of induced matching: instead of forcing the selected edges to induce only isolated edges, edge open packing allows them to induce disjoint stars.
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 set} 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 Brešar and Samadi concerning chordal graphs, we investigate the problem on three subclasses of chordal graphs. We give an \(O(n^2)\)-time algorithm for proper interval graphs, an \(O(n+m)\)-time algorithm for block graphs, where \(n=|V(G)|\) and \(m=|E(G)|\), and an \(O(n^3)\)-time algorithm for 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.