Geodesic
Fair Total Domination Number in Cactus Graphs
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 647-664.

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For k ≥ 1, a k-fair total dominating set (or just kFTD-set) in a graph G is a total dominating set S such that |N(v) ∩ S| = k for every vertex v ∈ V. The k-fair total domination number of G, denoted by ftdk(G), is the minimum cardinality of a kFTD-set. A fair total dominating set, abbreviated FTD-set, is a kFTD-set for some integer k ≥ 1. The fair total domination number of a nonempty graph G, denoted by ftd(G), of G is the minimum cardinality of an FTD-set in G. In this paper, we present upper bounds for the 1-fair total domination number of cactus graphs, and characterize cactus graphs achieving equality for the upper bounds.
Keywords: fair total domination, cactus graph 
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Hajian, Majid; Rad, Nader Jafari. Fair Total Domination Number in Cactus Graphs. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 647-664. http://geodesic.mathdoc.fr/item/DMGT_2021_41_2_a18/

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