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.

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In a graph, a vertex dominates itself and its neighbors. A subset S of vertices of a graph G is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number γ×2(G) of G is the minimum cardinality of a double dominating set of G. In this paper, we prove that the double domination number of a maximal outerplanar graph G of order n is bounded above by, where k is the number of pairs of consecutive vertices of degree two and with distance at least 3 on the outer cycle. We also prove that for a Hamiltonian maximal planar graph G of order n≥ 7.