Geodesic
Bounds on the total double Roman domination number of graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1033-1061

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Let G be a simple graph with no isolated vertex and let γ_tdR(G) be the total double Roman domination number of G. In this paper, we present lower and upper bounds on γ_tdR(G) of a graph G in terms of the order, open packing number and the numbers of support vertices and leaves, and we characterize all extremal graphs. We also prove that for any connected graph G of order n with minimum degree at least two, γ_tdR(G)≤⌊4n3⌋.
Keywords: double Roman domination, total double Roman domination, open packing 
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     title = {Bounds on the total double {Roman} domination number of graphs},
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Hao, Guoliang; Xie, Zhihong; Sheikholeslami, Seyed Mahmoud; Hajjari, Maryam. Bounds on the total double Roman domination number of graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1033-1061. http://geodesic.mathdoc.fr/item/DMGT_2023_43_4_a9/