Geodesic
On Grundy Total Domination Number in Product Graphs
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 225-247

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A longest sequence (v_1, . . ., v_k) of vertices of a graph G is a Grundy total dominating sequence of G if for all i, N(υ_i)\⋃_j=1^i-1N(υ_j)≠∅. The length k of the sequence is called the Grundy total domination number of G and denoted γ_gr^t(G). In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that γ_gr^t(G×H)≥γ_gr^t(G)γ_gr^t(H), conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic product we express γ_gr^t(G∘H) in terms of related invariant of the factors and find some explicit formulas for it. For the strong product, lower bounds on γ_gr^t(G⊠H) are proved as well as upper bounds for products of paths and cycles. For the Cartesian product we prove lower and upper bounds on the Grundy total domination number when factors are paths or cycles.
Keywords: total domination, Grundy total domination number, graph product 
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     title = {On {Grundy} {Total} {Domination} {Number} in {Product} {Graphs}},
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Brešar, Boštjan; Bujtás, Csilla; Gologranc, Tanja; Klavžar, Sandi; Košmrlj, Gašper; Marc, Tilen; Patkós, Balázs; Tuza, Zsolt; Vizer, Máté. On Grundy Total Domination Number in Product Graphs. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 225-247. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a14/