Double Roman domination number

Abstract

Given a graph G=(V,E), a function f:V→{0,1,2,3} having the property that if f(v)=0, then there exist v1,v2∈N(v) such that f(v1)=f(v2)=2 or there exists w∈N(v) such that f(w)=3, and if f(v)=1, then there exists w∈N(v) such that f(w)≥2 is called a double Roman dominating function (DRDF). The weight of a DRDF f is the sum f(V)=∑v∈Vf(v), and the minimum weight of a DRDF on G is the double Roman domination number, γdR(G) of G. In this paper, we show that γdR(G)+2⩽γdR(M(G))⩽γdR(G)+3, where M(G) is the Mycielskian graph of G. For any two positive integers a and b we construct a graph G and an induced subgraph H of G such that γdR(G)=a and γdR(H)=b and conclude that there is no relation between the double Roman domination number of a graph and its induced subgraph. We also study the impact of edge addition on double Roman domination number and find an upperbound in terms of order and diameter.