On Total Double Italian Domination in Graphs

Abstract

For a simple graph \(G = \left(V\left(G\right), E\left(G\right)\right)\), a total double Italian dominating function is a function \(f: V(G) \to \{0,1,2,3\}\) with the properties that for every vertex \(v \in V\left(G\right)\) with \(f\left(v\right) \in \{0,1\}\), \(\sum_{u \in N\left[v\right]} f\left(u\right) \geq 3\), and every vertex \(v \in V(G)\) with \(f(v) \neq 0\) has a neighbor \(u\) with \(f(u) \neq 0\). 

The weight of a total double Italian dominating function is the sum 
\[
\omega_G\left(f\right) = \sum_{v \in V\left(G\right)} f\left(v\right),
\]
and the minimum weight of all total double Italian dominating functions on a graph \(G\) is the total double Italian domination number, denoted by \(\gamma_{tdI}\left(G\right)\).

In this paper, we explore further the concept of total double Italian domination. We characterize graphs \(G\) with smaller values for \(\gamma_{tdI}(G)\). Additionally, we characterize the total double Italian dominating function on the join, corona, edge corona, and complementary prism of graphs. Exact values or bounds are also determined for their respective total double Italian domination numbers.