A note on the double Roman domination number of graphs

summary:For a graph $G=(V,E)$, a double Roman dominating function is a function $f\colon V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two neighbors assigned $2$ under $f$ or one neighbor with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must have at least one neighbor with $f(w)\geq 2$. The weight of a double Roman dominating function $f$ is the sum $f(V)=\sum \nolimits _{v\in V}f(v)$. The minimum weight of a double Roman dominating function on $G$ is called the double Roman domination number of $G$ and is denoted by $\gamma _{\rm dR}(G)$. In this paper, we establish a new upper bound on the double Roman domination number of graphs. We prove that every connected graph $G$ with minimum degree at least two and $G\neq C_{5}$ satisfies the inequality $\gamma _{\rm dR}(G)\leq \lfloor \frac {13}{11}n\rfloor$. One open question posed by R. A. Beeler et al. has been settled

Characterization of graphs with equal domination and connected domination numbers

AbstractArumugam and Paulraj Joseph (Discrete Math 206 (1999) 45) have characterized trees, unicyclic graphs and cubic graphs with equal domination and connected domination numbers. In this paper, we extend their results and characterize the class of block graphs and cactus graphs for which the domination number is equal to the connected domination number