دوستعلی مژده

We continue the study of restrained double Roman domination in graphs. For a graph G = V (G ) , E(G ) , a double Roman dominating function fis called a restrained double Roman dominating function (RDRD function) if the subgraph induced by { v ∈ V (G ) | f(v ) = 0 } has no isolated vertices. The restrained double Roman domination number (RDRD number) γrdR (G ) is the minimum weight  v ∈ V(G ) f(v ) taken over all RDRD functions of G . We first prove that the problem of computing γrdR is NP-hard even for planar graphs, but it is solvable in linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary graphs. Relationships between γrdR and some well- known parameters such as restrained domination number γr , domination number γor re- strained Roman domination number γrR are investigated in this paper by bounding γrdR from below and above involving γr , γor γrR for general graphs, respectively. We prove that γrdR (T ) ≥n + 2 for any tree T  = K 1 ,n −1 of order n ≥2 and characterize the family of all trees attaining the lower bound. The characterization of graphs with small RDRD num- bers is given in this paper.