An upper bound on the total double Roman domination number

Source abstract

Let $G=(V,E)$ be a graph with no isolated vertex. A total double Roman dominating function (TDRDF) is a DRDF such that the vertices of positive weight under function are not isolated. In this paper, we initiate to study the bound of TDRD number of a graph $G$ from above, and we improve this bound. In particular, we show that $\gamma_{tdR}(G)\le \frac{4n}{3}$, whenever $G$ is a graph of order $n$ and minimum degree at least $2$. Received: 16 May 2021 | Revised: 12 February 2025 | Accepted: 03 September 2025