No abstract

An outer-independent Italian dominating function (OIIDF) on a graph G is a function f : V (G) −→ {0, 1, 2} such that every vertex v ∈ V (G) with f (v) = 0 has at least two neighbors assigned 1 under f or one neighbor w with f (w) = 2, and the set {u ∈ V (G)|f (u) = 0} is independent. An outer-independent double Italian dominating function (OIDIDF) on a graph G is a functionThe minimum weight of an OIIDF (respectively, OIDIDF) on a graph G is called the outer-independent Italian (respectively, outer-independent double Italian) domination number of G. We characterize all trees T with outer-independent double Italian domination number twice the outer-independent Italian domination number. We also present lower bounds on the outer-independent double Italian domination number of a connected graph G in terms of the order, minimum and maximum degrees.