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Edge open packing on subclasses of chordal graphs

Kamal Santra

TL;DR

The paper addresses the Maximum Edge Open Packing Problem (MEOP) on chordal graphs by deriving efficient algorithms for three subclasses. It introduces a polynomial-time dynamic-programming framework for proper interval graphs based on a bi-compatible elimination ordering, a rooted-cut-tree approach for block graphs with detailed subproblem recurrences, and a linear-time method for split graphs leveraging clique–independent set structure. Together, these results advance the tractability of edge open packing in chordal graphs and partially answer an open question in the literature. The work has implications for related coloring and packing problems and provides practical algorithms for structural graph classes common in applications.

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 $ρ_{e}^o(G)$. 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}.

Edge open packing on subclasses of chordal graphs

TL;DR

The paper addresses the Maximum Edge Open Packing Problem (MEOP) on chordal graphs by deriving efficient algorithms for three subclasses. It introduces a polynomial-time dynamic-programming framework for proper interval graphs based on a bi-compatible elimination ordering, a rooted-cut-tree approach for block graphs with detailed subproblem recurrences, and a linear-time method for split graphs leveraging clique–independent set structure. Together, these results advance the tractability of edge open packing in chordal graphs and partially answer an open question in the literature. The work has implications for related coloring and packing problems and provides practical algorithms for structural graph classes common in applications.

Abstract

Let be a graph where and are the vertex and edge sets, respectively. In a graph , two edges are said to have a \emph{common edge} if joins an endpoint of to an endpoint of in . A subset is called an \emph{edge open packing set} in if no two edges in share a common edge in , and the largest size of such a set in is known as the \emph{edge open packing number}, represented by . 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}.
Paper Structure (6 sections, 9 theorems, 6 equations, 1 figure, 2 algorithms)

This paper contains 6 sections, 9 theorems, 6 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. EOP set in proper interval graphs
  4. EOP set in block graphs
  5. EOP set in split graphs
  6. Conclusion

Key Result

Lemma 4

Let G be a connected proper interval graph with a BCO $\sigma=\{v_1, v_2, \ldots, v_n\}$. Then Moreover, $\rho_1\left(G\right)=\max \left\{\rho'\left(G\right), \rho"\left(G\right)\right\}$.

Figures (1)

  • Figure 1: On the right is a block graph $G$, in the middle is its cut-tree$T_G^{v_1}$, and on the left side are the graphs $G_v$ and $G_v^B$, respectively

Theorems & Definitions (16)

  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • Lemma 10
  • ...and 6 more