Geodesic
Total Domination in Generalized Prisms and a New Domination Invariant
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1165-1178

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In this paper we complement recent studies on the total domination of prisms by considering generalized prisms, i.e., Cartesian products of an arbitrary graph and a complete graph. By introducing a new domination invariant on a graph G, called the k-rainbow total domination number and denoted by γkrt(G), it is shown that the problem of finding the total domination number of a generalized prism G □ Kk is equivalent to an optimization problem of assigning subsets of 1, 2, . . ., k to vertices of G. Various properties of the new domination invariant are presented, including, inter alia, that γkrt(G) = n for a nontrivial graph G of order n as soon as k ≥ 2Δ(G). To prove the mentioned result as well as the closed formulas for the k-rainbow total domination number of paths and cycles for every k, a new weight-redistribution method is introduced, which serves as an efficient tool for establishing a lower bound for a domination invariant.
Keywords: domination, k -rainbow total domination, total domination 
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     title = {Total {Domination} in {Generalized} {Prisms} and a {New} {Domination} {Invariant}},
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Tepeh, Aleksandra. Total Domination in Generalized Prisms and a New Domination Invariant. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1165-1178. http://geodesic.mathdoc.fr/item/DMGT_2021_41_4_a19/