Construction of trees with unique minimum semipaired dominating sets

Abstract

Let $G$ be a graph with vertex set $V$ and no isolated vertices. A subset $S\subseteq V$ is a semipaired dominating set of $G$ if every vertex in $V\setminus S$ is adjacent to a vertex in $S$ and $S$ can be partitioned into two-element subsets such that the vertices in each subset are at most distance two apart. We present a method of building trees having a unique minimum semipaired dominating set.