Edge open packing on subclasses of chordal graphs

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 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 $\rho_{e}^o(G)$. The \textsc{Maximum Edge Open Packing Problem} is to find an edge open packing set of a given graph with maximum size. In [Bre{\v{s}}ar and Samadi. Edge open packing: complexity, algorithmic aspects, and bounds. Theor. Comput. Sci., 2024.], Bre{\v{s}}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}.