On the outer-independent double Italian domination number

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 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.