Some Results On(Independent)Roman {2}-domination Number And Double Roman Domination Number

Let G=(V,E)be a graph with the vertex set V=V(G)and the edge set E=E(G).A function f:V→ {0,1,2} is a Roman {2}-dominating function on a graph G if f satisfies the following condition:if f(v)=0,then there is v1,v2∈N(v)such that f(u1)=f(v2)=1 or there is u∈ N(v)such that f(u)=2.Let Vif denote the set of vertices assigned i by function f.The weight of a Roman {2}-dominating function f is the sum f(V)=∑v∈V f(V).The Roman {2}-domination number is the minimum weight of a Roman {2}-dominating function on G,denot-ed by γ{R2}(G).A Roman {2}-dominating function on G with weight γ{R2}(G)is called a γ{R2}-function of G.A function f:V→{0,1,2} is an independent Ro-man {2}-dominating function on a graph G if f is a Roman {2}-dominating func-tion and V1f ∩ V2f is an independent set.In this paper,we study the relationship of y{R2}(M(G))and γ{R2}(G),where M(G)is the Mycielskian graph of G.For two positive integer a and b,where a>2,b≥2,we costruct a graph G and an induced subgraph H of G such that γ{R2}(G)=a and γ{R2}(H)=b.Moreover,we study the impact of edge addition on the Roman {2}-domination number;In independent Roman {2}-domination,we give a bound on independent Roman{2}-domination number for the proper interval graphs,and propose a linear time algorithm for computing the independent Roman {2}-domination number of G,where G is a single cycle graph with rest,riction.A function f:V → {0,1,2,3} is a double Roman dominating function on a graph G if f satisfies the following conditions:if f(v)=0,then there is v1,v2∈N(v)such that f(v1)=f(v2)=2,or there is u ∈N(v)such that f(u)=3;and if f(v)=1,then there is u∈N(v)such that f(u)=3.Simi-lar to the relevant definitions of Roman{2}-domination,we can give the relevant definitions of independent Roman{2}-domination and double Roman domination respectively.In double Roman domination,we discuss the uniqueness of the dou-ble Roman dominating functions with the minimum weight on a path.