bounds on the outer-independent double italian domination number

An outer-independent double italian dominating function (oididf) on a graph g with vertex set v(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 oididf f is the value w(f)=∑v∈v(g)f(v). the minimum weight of an oididf on a graph g is called the outer-independent double italian domination number γoidi(g) of g. we present sharp lower bounds for the outer-independent double italian domination number of a tree in terms of diameter, vertex covering number and the order of the tree.