AUTHOR=Rao Yongsheng , Kosari Saeed , Sheikholeslami Seyed Mahmoud , Chellali M. , Kheibari Mahla 

TITLE=On the Outer-Independent Double Roman Domination of Graphs

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 6 - 2020

YEAR=2021

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2020.559132

DOI=10.3389/fams.2020.559132

ISSN=2297-4687

ABSTRACT=‎An outer-independent double Roman dominating function (OIDRDF) of a graph $G$‎ ‎is a function $h:V(G)\rightarrow \{0,1,2,3\}$ such that \textrm{(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 $2,$ \textrm{(ii) }every vertex $v$ with‎
‎$f(v)=1$ is adjacent to at least one vertex {with label greater than 1‎, }% ‎\textrm{(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 $\gamma _{oidR}(G)$ is the‎ ‎minimum weight of an OIDRDF on $G$‎. ‎Abdollahzadeh Ahangar et al‎. ‎(2020)‎ ‎showed that for any tree $T$\ of order $n\geq 3$\textbf{, ‎}$\gamma‎_{oidR}(T)\leq 5n/4,$ 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, ‎$\gamma _{oidR}(G)\leq \frac{4n}{3%‎‎}$‎.