Study On Global Double Roman Domination Number In Graphs

Given a graph G=(V(G),E(G)),a function f:V(G)→ {0,1,2,3} having the property that(i)if f(v)=0,then there exists v1,v2∈N(v)such that f(v1)=f(v2)=2 or there exists w∈N(v)such that f(w)=3,(ii)if f(v)=1,then there exists u∈N(v)such that f(u)≥ 2 is called a double Roman dominating function(DRDF).A DRDF f:V(G)→ {0,1,2,3} is called a global double Roman domination function(GDRDF)if f is also a double Roman dominating function of the complete G.The weight w(f)of a GDRDF f is the sum w(f)=∑v∈Vf(v),and the minimum weight of a GDRDF on G is called the global double Roman domination number of G and is denoted by γqdR(G).We say that a function f of G is a γqdR-function if it is a GDRDF and w(f)=γqdR(G).In this paper,we first study the upper and lower bound problem of the global double Roman domination number of the graph G.By analyzing the three pa-rameters of diameter,girth and degree of the graph,we present the relationship between the upper and lower bound problem of the global double Roman domi-nation number of the graph G and these three parameters,and some of them is proved to be tight.Next,an open question that characterize the graphs G withγgdR(G)=γdR(G)+t for each t ∈{0,1,2,3,4,5} posed by Z.Shao et al.(2019)in global double Roman domination in graphs.We study the case of the open ques-tion in the tree.We prove that for any trees T of order n≥ 4,γgdR(T)≤γdR(T)+3 and characterize all trees with γgdR(T)=γdR(T)+3,γgdR(T)=γdR(T)+2 andγgdn(T)=γdR(T)+1.