<![CDATA[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 function f : V(G)→{0, 1, 2, 3} such that if f(v)∈{0, 1} for a vertex v ∈ V(G), then ∑u ∈ N[v]f(u)≥3 and the set {u ∈ V(G)|f(u)=0} is independent. The weight of an OIIDF (respectively, OIDIDF) f is the value w(f)=∑v ∈ V(G)f(v). The 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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