AN UPPER BOUND ON THE TOTAL OUTER-INDEPENDENT DOMINATION NUMBER OF A TREE

A total outer-independent dominating set of a graph G = (V (G); E (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 total outer-independent domination number of a graph G, denoted by rt(oi) (G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n >= 4, with l leaves and s support vertices we have rt(oi) (T) <= (2 n + s - l)/3, and we characterize the trees attaining this upper bound.