Vertices in all minimum paired-dominating sets of block graphs

TL;DR

This paper introduces a linear-time algorithm to identify vertices that are part of all minimum paired-dominating sets in block graphs, advancing understanding of their structural properties.

Contribution

It provides the first efficient algorithm to determine vertices in all minimum paired-dominating sets of block graphs.

Findings

Algorithm runs in linear time.

Identifies vertices in all minimum paired-dominating sets.

Enhances structural understanding of block graphs.

Abstract

Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$ is a paired-dominating set if every vertex in $V-S$ has at least one neighbor in $S$ and the subgraph induced by $S$ contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.