Optimal Algorithm for Paired-Domination in Distance-Hereditary Graphs

TL;DR

This paper presents a new linear-time algorithm for the paired-domination problem in distance-hereditary graphs, improving upon previous quadratic solutions and leveraging graph decomposition for further efficiency.

Contribution

It introduces an $O(n+m)$-time algorithm for paired-domination in distance-hereditary graphs, with an optimized $O(n)$-time version using a decomposition tree.

Findings

The algorithm runs in linear time for general distance-hereditary graphs.

Using a decomposition tree reduces the algorithm's complexity to linear time.

The approach significantly improves the efficiency over previous methods.

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 $G$.