On the Outer-Independent Double Roman Domination of Graphs - Rao, Yongsheng; Kosari, Saeed; Sheikholeslami, Seyed Mahmoud; Chellali, M.; Kheibari, Mahla (Frontiers in Applied Mathematics and Statistics, 2021)
‎An outer-independent double Roman dominating function (OIDRDF) of a graph $G$‎ ‎is a function $h:V(G)→ \{0,1,2,3\}$ such that (i) every‎ ‎vertex $v$ with $f(v)=0$ is adjacent to at least one vertex with label $3$‎ ‎or to least two vertices with label $2,$ (ii) every vertex $v$ with‎ ‎$f(v)=1$ is adjacent to at least one vertex {with label greater than 1‎, }% ‎(iii) all vertices labeled by $0$ {is an independent set}‎. ‎The‎ ‎weight of an OIDRDF is the sum of its function values over all vertices‎. ‎The‎ ‎outer-independent double Roman domination number $γ _{oidR}(G)$ is the‎ ‎minimum weight of an OIDRDF on $G$‎. ‎Abdollahzadeh Ahangar et al‎. ‎(2020)‎ ‎showed that for any tree $T$\ of order $n≥ 3$, ‎$γ‎_{oidR}(T)≤ 5n/4,$ and posed the problem of characterizing those trees‎ ‎attaining equality‎. ‎In this paper‎, ‎we solve this problem and we‎ ‎give additional bounds on the outer-independent double Roman domination‎ ‎number‎. ‎In particular‎, ‎we show that for any connected graph $G$‎ ‎of order $n$ with minimum degree at least two in which the set of vertices‎ ‎with degree at least three is independent‎, ‎$γ _{oidR}(G)≤ \frac{4n}{3%‎‎}$‎.