On the Outer-Independent Double Roman Domination of Graphs

Yongsheng Rao,Seyed Mahmoud Sheikholeslami,Saeed Kosari,Mahla Kheibari,M Chellali

doi:10.3389/fams.2020.559132

Abstract

An outer-independent double Roman dominating function (OIDRDF) of a graphGis a functionh:V(G)→{0,1,2,3}such that i) every vertexvwithf(v)=0is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertexvwithf(v)=1is adjacent to at least one vertex with label greater than 1, and iii) all vertices labeled by 0 are an independent set. The weight of an OIDRDF is the sum of its function values over all vertices. The outer-independent double Roman domination numberγoidR(G) is the minimum weight of an OIDRDF onG. It has been shown that for any treeTof ordern≥ 3,γoidR(T) ≤ 5n/4 and the problem of characterizing those trees attaining equality was raised. In this article, we solve this problem and we give additional bounds on the outer-independent double Roman domination number. In particular, we show that, for any connected graphGof ordernwith minimum degree at least two in which the set of vertices with degree at least three is independent,γoidR(T) ≤ 4n/3.

Highlights

We consider only simple connected graphs G with vertex set V V(G) and edge set E E(G), where n |V| is the order of G
We begin by recalling the question, posed in Ref. 19, on whether the 5n/4 upper bound on the OIDRD-number for trees remains valid for arbitrary graphs
We continued the study of outer-independent double Roman domination number and we characterized the trees T of order n ≥ 3, for which coidR(T) ≤ 5n/4, answering a problem posed by Abdollahzadeh Ahangar et al, Ref. 19

Summary

INTRODUCTION

We consider only simple connected graphs G with vertex set V V(G) and edge set E E(G), where n |V| is the order of G. An outerindependent double Roman dominating function (OIDRDfunction) of a graph G is a DRDF h such that the set of vertices assigned a 0 under h is independent. Mojdeh et al, Ref. 20, proved that the decision problem associated with coidR (G) is NP-complete even when restricted to planar graphs with maximum degree at most four They characterized the families of all connected graphs with small outer-independent double Roman domination numbers. We prove that, for any connected graph G of order n with minimum degree at least two in which the set of vertices with degree at least three is independent, coidR (G) ≤ 4n/3. Any coidR(T′′)-function f can be extended to an OIDRD-function of T by assigning a 3 to v3, 2 to v1, and a 0 to v2, w. V5 has a neighbor with weight at least two and the function h′ defined on V(T) by h′(v3) 3, h′(v1) 2, h′(v5) 1, h′(v4) h′(w) h′(v2) 0, and h′(x) f (x) for x ∈ V(T′′) − {v4, v5} is an OIDRD-function of T of weight ω(h′) + 4 5(n − 4)/4 + 4 < 5n/4, a contradiction

SLIGHTLY IMPROVED BOUNDS FOR TREES

GRAPHS WITH MINIMUM DEGREE TWO

CONCLUSION

DATA AVAILABILITY STATEMENT