Quadruple Roman Domination in Trees

Zheng Kou,Guoliang Hao,Jafar Amjadi,Nesa Khalili,Saeed Kosari

doi:10.3390/sym13081318

Abstract

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

Highlights

We present bounds on γ[4R] ( T ) in terms of (Roman) domination number and we give a characterization of extremal trees
We first prove that 4γ( T ) + 1 ≤ γ[4R] ( T ) ≤ 4γR ( T ) − 3 and we present a characterization of trees for which γ[4R] ( T ) = 4γ( T ) + 1 and γ[4R] ( T ) + 3 =
We focused on trees and we presented lower and upper bounds on the quadruple Roman domination number of trees and characterized all extremal trees

Summary

Introduction

For notation and graph theory terminology, we in general follow Haynes et al [1]. Let. G be a simple graph with vertex set V ( G ) and edge set E( G ) and let n = n( G ) = |V ( G )|. The open neighborhood of u is the set N (u) = { x ∈ V ( G ) | ux ∈ E( G )}. The closed neighborhood of u is the set N [u] = {u} ∪ N (u). Let d(u) = | N (u)| denote the degree of a vertex u of G. The number of vertices in distance 2 of u is denoted by N2 (u)

Objectives

Conclusion