Double Roman Domination in Graphs with Minimum Degree at Least Two and No $C_{5}$-cycle

Abstract

A double Roman dominating function (DRDF) on a graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with $f(w)=3$, and if $f(v)=1$, then vertex v must have at least one neighbor w with $f(w)\ge 2$. The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number $\gamma _{dR}(G)$ is the minimum weight of a DRDF on G. Recently, Khoeilar et al. (Discrete Appl Math 270:159–167, 2019) proved that a connected graph G of order n with minimum degree two different from $C_{5}$ and $C_{7}$, satisfies $\gamma _{dR}(G)\le \frac{11}{10}n.$ In this paper, we show that this upper bound can be improved to $\frac{20}{19}n$ if G is restricted to connected graphs of order n with minimum degree at least two and no $C_{5}$-cycle such that $G\notin \{C_{7},C_{11},C_{13},C_{17}\}.$ Moreover, we provide an infinite family of graphs attaining this bound.