From Italian domination in lexicographic product graphs to w-domination in graphs

Identifier:  imarina:9220190

Handle: https://hdl.handle.net/20.500.11797/imarina9220190

Authors:  Cabrera Martinez, Abel; Estrada-Moreno, Alejandro; Alberto Rodriguez-Velazquez, Juan

Abstract:
In this paper, we show that the Italian domination number of every lexicographic product graph G o H can be expressed in terms of five different domination parameters of G. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let N(v) denote the open neighbourhood of v ∈ V (G), and let w = (w0,w1,⋯, wl) be a vector of nonnegative integers such that w0≥ 1. We say that a function f : V (G) → {0, 1,⋯, l} is a w-dominating function if f(N(v)) = Σu∈N(v)f(u) ≥ wifor every vertex v with f(v) = i. The weight of f is defined to be ω(f) = Σv∈V (G)f(v). The w-domination number of G, denoted by γw(G), is the minimum weight among all w-dominating functions on G. Specifically, we show that γI(G 0 H) = γw(G), where w ∈ {2} × {0, 1, 2}land l ∈ {2, 3}. The decision on whether the equality holds for specific values of w0,⋯, wlwill depend on the value of the domination number of H. This paper also provides preliminary results on γw(G) and raises the challenge of conducting a detailed study of the topic.