Trees with Double Roman Domination Number Twice the Domination Number Plus Two

Abstract

A double Roman dominating function (DRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ such that (i) every vertex v with $f(v)=0$ is adjacent to at least two vertices assigned a 2 or to at least one vertex assigned a 3,  (ii) every vertex v with $f(v)=1$ is adjacent to at least one vertex w with $f(w)\ge 2.$ The weight of a DRDF is the sum of its function values over all vertices. The double Roman domination number $\gamma _{\rm dR}(G)$ equals the minimum weight of a double Roman dominating function on G. Beeler, Haynes and Hedetniemi showed that for every non-trivial tree T, $\gamma _{\rm dR}(T)\ge 2\gamma (T)+1,$ where $\gamma (T)$ is the domination number of T. A characterization of extremal trees attaining this bound was given by three of us. In this paper, we characterize all trees T with $\gamma _{\rm dR}(T)=2\gamma (T)+2$.