Double Roman Domination

Problem Definition

For a graph $G = (V, E)$

a double Roman dominating function is a function

$f : V \to {0, 1, 2, 3}$

having the property that

if $f (v) = 0$ ,
- then vertex $v$ must have at least two neighbours assigned $f (w) = 2$ under $f$ or one neighbor with $f (w) = 3$ ,
if $f (v) = 1$ ,
- then vertex $v$ must have at least one neighbor with $f (w) \geq 2$ .
The weight( $w$ ) of a double Roman dominating function is the sum $\sum_{v ϵ_{V}} f (v)$ , and the minimum weight of a double Roman dominating function on is the double Roman domination number of .