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 $$\gamma _{\rm dR}(T)=2\gamma (T)+2$$ .