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 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.

Outer independent total double Italian domination number

“…In what follows we establish some lower bounds on γ oi tdR (G). The proof of the next result is similar to the proof of Theorem 3.1 in [1].…”

“…An outer independent double Italian dominating function (OIDIDF) of a graph G is a DIDF for which the vertices with weight 0 are independent. The outer independent double Italian domination number γ oidI (G) is the minimum weight of an OIDIDF on G (see [1], [3], [4], [19]). An OIDIDF on G with weight γ oidI (G) is called a γ oidI (G)-function.…”