A Note on the Italian Domination Number and Double Roman Domination Number in Graphs

Abstract

An Italian dominating function (IDF) on a graph G = (V,E) is a function f: V → {0,1,2} satisfying the property that for every vertex $v\in V$, with f(v) = 0, $\sum _{u\in N_{(v)}}f(u)\ge 2$. The weight of an IDF f is the value $w(f)=f(V)=\sum _{u\in V}f(u)$. The minimum weight of an IDF on a graph G is called the Italian domination number of G, denoted by ${\gamma }_I(G)$. For a graph G = (V,E), a double Roman dominating function (or just DRDF) is a function f: V → {0, 1, 2,3} having the property that if $f(v)=O$ for a vertex $u$, then $u$ has at least two neighbors assigned 2 under for one neighbor assigned 3 under f, and if $f(v)=1$, then u has at least one neighbor with f(w) ≥ 2. The weight of a DRDF f is the sum $f(V)=\sum _{u\in V}f(v)$, and the minimum weight of a DRDF on G is the double Roman domination number oi G, denoted by ${\gamma }_{dR}(G)$. In this paper we show that ${\gamma }_{d}R(G)/2\le {\gamma }_I(G)\le 2{\gamma }_{d}R(G)/3$, and characterize all trees T with ${\gamma }_I(T)=2{\gamma }_{d}R(T)/3$.