Labelling Algorithms for Paired-domination Problems in Block and Interval Graphs

TL;DR

This paper introduces two linear-time algorithms for finding minimum paired-dominating sets in block and interval graphs, correcting previous misconceptions and establishing NP-completeness in other graph classes.

Contribution

The paper presents novel linear-time algorithms for paired-domination in block and interval graphs and clarifies the problem's computational complexity in other graph types.

Findings

Linear-time algorithms for block and interval graphs

Correction of previous algorithmic misconceptions

NP-completeness in bipartite, chordal, and split graphs

Abstract

Let $G=(V,E)$ be a graph without isolated vertices. A set $S\subseteq V$ is a paired-domination set if every vertex in $V-S$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng [ Paired domination on interval and circular-arc graphs, Disc. Appl. Math. 155(2007),2077-2086], we present two linear time algorithms to find a minimum cardinality paired-dominating set in block and interval graphs. In addition, we prove that paired-domination problem is {\em NP}-complete for bipartite graphs, chordal graphs, even split graphs.