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 v 1 , v 2 ∈ N ( v ) such that f ( v 1 ) = f ( v 2 ) = 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 ∈ V f ( v ) , and the minimum weight of a DRDF on G is the double Roman domination number, γ d R ( G ) of G . In this paper, we show that γ d R ( G ) + 2 ⩽ γ d R ( M ( G ) ) ⩽ γ d R ( 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 γ d R ( G ) = a and γ d R ( 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.