On Trees with Double Domination Number Equal to the 2-Outer-Independent Domination Number Plus One

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 double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2-dominating set if every vertex of V (G) \ D has at least two neighbors in D, while it is a 2-outer-independent dominating set of G if additionally the set V (G)\D is independent. The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G. This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.