### Closed formulas for the total Roman domination number of lexicographic product graphs

#### Abstract

Let *G* be a graph with no isolated vertex and *f*: *V*(*G*) → {0, 1, 2} a function. Let *V*_{i} = {*x* ∈ *V*(*G*) : *f*(*x*) = *i*} for every *i* ∈ {0, 1, 2}. We say that *f* is a total Roman dominating function on *G* if every vertex in *V*_{0} is adjacent to at least one vertex in *V*_{2} and the subgraph induced by *V*_{1} ∪ *V*_{2} has no isolated vertex. The weight of *f* is *ω*(*f*) = ∑_{v ∈ V(G)}*f*(*v*). The minimum weight among all total Roman dominating functions on *G* is the total Roman domination number of *G*, denoted by *γ*_{tR}(*G*). It is known that the general problem of computing *γ*_{tR}(*G*) is NP-hard. In this paper, we show that if *G* is a graph with no isolated vertex and *H* is a nontrivial graph, then the total Roman domination number of the lexicographic product graph *G* ∘ *H* is given by *γ*_{tR}(*G* ∘ *H*) = 2*γ*_{t}(*G*) if *γ*(*H*) ≥ 2, and *γ*_{tR}(*G* ∘ *H*) = *ξ*(*G*) if *γ*(*H*) = 1, where *γ*(*H*) is the domination number of *H*, *γ*_{t}(*G*) is the total domination number of *G* and *ξ*(*G*) is a domination parameter defined on *G*.

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MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2284.aeb

ISSN: 1855-3974

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