Perfect double Roman domination of trees

Abstract

For a graph G with vertex set V(G) and function f:V(G)→{0,1,2,3}, let Vi be the set of vertices assigned i by f. A perfect double Roman dominating function of a graph G is a function f:V(G)→{0,1,2,3} satisfying the conditions that (i) if u∈V0, then u is either adjacent to exactly two vertices in V2 and no vertex in V3 or adjacent to exactly one vertex in V3 and no vertex in V2; and (ii) if u∈V1, then u is adjacent to exactly one vertex in V2 and no vertex in V3. The perfect double Roman domination number of G, denoted γdRp(G), is the minimum weight of a perfect double Roman dominating function of G. We prove that if T is a tree of order n≥3, then γdRp(T)≤9n∕7. In addition, we give a family of trees T of order n for which γdRp(T) approaches this upper bound as n goes to infinity.