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Optimal Algorithm for Paired-Domination in Distance-Hereditary Graphs

Ta-Yu Mu, Ching-Chi Lin

TL;DR

The paper addresses the minimum paired-domination problem on distance-hereditary graphs and presents an optimal $O(n+m)$-time algorithm, with a further $O(n)$-time improvement when a decomposition tree of $G$ is provided. The core approach combines a linear-time decomposition-tree construction with a bottom-up dynamic programming framework that tracks carefully defined subproblem parameters through recursive graph-assembly operations (true twin, false twin, and attachment). Key contributions include two recurrences (Eq. 1 and Eq. 2) and the introduction of auxiliary quantities such as $oldsymbol{ umy{min}}(v)$, $oldsymbol{ umy{ he alpha}}(v)$, $oldsymbol{ umy{ he beta}}(v)$, and the mty indicators, all computed in $O(1)$ per internal node. The results yield a linear-time algorithm for the minimum paired-dominating set on distance-hereditary graphs, with potential extensions to circle- and planar-graph families through related structural decompositions and approximation strategies. This work significantly improves prior $O(n^2)$ bounds and provides a framework for efficient exact solutions in structured graph classes, with practical implications for security and surveillance applications that rely on paired-dominating configurations.

Abstract

The domination problem and its variants represent a classical domain within algorithmic graph theory. Among these variants, the paired-domination problem holds particular prominence due to its real-world implications in security and surveillance domains. Given an input graph $G$, the paired-domination problem involves identifying a minimum dominating set $D$ that induces a subgraph of $G$ with a perfect matching. Lin et al.~[\emph{Paired-domination problem on distance-hereditary graphs}, Algorithmica, 2020] previously presented a solution to this problem with a time complexity of $O(n^2)$. This paper significantly enhances their findings by introducing an $O(n+m)$-time algorithm. Furthermore, the time complexity of this algorithm can be reduced to $O(n)$ when provided with a decomposition tree for the graph $G$.

Optimal Algorithm for Paired-Domination in Distance-Hereditary Graphs

TL;DR

The paper addresses the minimum paired-domination problem on distance-hereditary graphs and presents an optimal -time algorithm, with a further -time improvement when a decomposition tree of is provided. The core approach combines a linear-time decomposition-tree construction with a bottom-up dynamic programming framework that tracks carefully defined subproblem parameters through recursive graph-assembly operations (true twin, false twin, and attachment). Key contributions include two recurrences (Eq. 1 and Eq. 2) and the introduction of auxiliary quantities such as , , , and the mty indicators, all computed in per internal node. The results yield a linear-time algorithm for the minimum paired-dominating set on distance-hereditary graphs, with potential extensions to circle- and planar-graph families through related structural decompositions and approximation strategies. This work significantly improves prior bounds and provides a framework for efficient exact solutions in structured graph classes, with practical implications for security and surveillance applications that rely on paired-dominating configurations.

Abstract

The domination problem and its variants represent a classical domain within algorithmic graph theory. Among these variants, the paired-domination problem holds particular prominence due to its real-world implications in security and surveillance domains. Given an input graph , the paired-domination problem involves identifying a minimum dominating set that induces a subgraph of with a perfect matching. Lin et al.~[\emph{Paired-domination problem on distance-hereditary graphs}, Algorithmica, 2020] previously presented a solution to this problem with a time complexity of . This paper significantly enhances their findings by introducing an -time algorithm. Furthermore, the time complexity of this algorithm can be reduced to when provided with a decomposition tree for the graph .

Paper Structure

This paper contains 26 sections, 1 theorem, 13 equations, 2 figures, 9 algorithms.

Table of Contents

  1. Introduction
  2. Algorithm for Distance-Hereditary Graphs
  3. Decomposition Trees
  4. The Algorithm
  5. Correctness and Implementation Details
  6. Key properties of $\hat{\gamma}_k(v)$
  7. True Twin Operation $(G=G_l \otimes G_r)$
  8. Determine $\hat{min}(v)$, $\hat{\alpha}(v)$, and $\hat{\beta}(v)$ for true twin operation
  9. Equation (\ref{['eq2']}) holds for true twin operation $\otimes$
  10. False Twin Operation $(G=G_l \odot G_r)$
  11. Determine $\hat{min}(v)$, $\hat{\alpha}(v)$, and $\hat{\beta}(v)$ for false twin operation
  12. Equation (\ref{['eq2']}) holds for false twin operation $\odot$
  13. Attachment Operation $(G=G_l \oplus G_r)$
  14. Case $C_1$: conditions $D_1$ and $D_2$ are false
  15. Determine $\hat{min}(v)$, $\hat{\alpha}(v)$, and $\hat{\beta}(v)$ for case $C_1$
  16. ...and 11 more sections

Key Result

theorem thmcountertheorem

Given a distance-hereditary graph $G$, a minimum paired-dominating set of $G$ can be determined in $O(n + m)$ time.

Figures (2)

  • Figure 1: A distance-hereditary graph $G$ with its corresponding decomposition tree $T$ rooted at $\kappa$.
  • Figure 2: The relationship between the unpaired vertex numbers $k$ and their corresponding $\hat{\gamma}_k(v)$ of $\hat{G}(v)$.

Theorems & Definitions (69)

  • proof
  • proof
  • theorem thmcountertheorem
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 59 more