On the Ratio Between 2-Domination and Total Outer-Independent Domination Numbers of Trees

A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V (G) | D has at least two neighbors in D. A total outer-independent 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, and the set V (G) | D is independent. The 2-domination (total outer-independent domination, respectively) number of a graph G is the minimum cardinality of a 2-dominating (total outer-independent dominating, respectively) set of G. We investigate the ratio between 2-domination and total outer-independent domination numbers of trees.