On the outer-independent double Italian domination number - Noor A'lawiah Abd Aziz; Hailiza Kamarulhaili; Farzaneh Azvin; Nader Jafari Rad (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2022)
‎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.