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)≤ 4n3%‎‎‎.