Vertices in all minimum paired-dominating sets of block graphs

Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$isapaired-dominatingsetifeveryvertexin$V-S$ 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.