On Trees with Double Domination Number Equal to Total Domination Number Plus One

Abstract

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ is dominated by at least two vertices of $D$. The total (double, respectively) domination number of a graph $G$ is the minimum cardinality of a total (double, respectively) dominating set of $G$. We characterize all trees with double domination number equal to total domination number plus one.