Geodesic
Fair Domination Number in Cactus Graphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 489-503.

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For k ≥ 1, a k-fair dominating set (or just kFD-set) in a graph G is a dominating set S such that |N(v) ∩ S| = k for every vertex v ∈ V S. The k-fair domination number of G, denoted by fdk(G), is the minimum cardinality of a kFD-set. A fair dominating set, abbreviated FD-set, is a kFD-set for some integer k ≥ 1. The fair domination number, denoted by fd(G), of G that is not the empty graph, is the minimum cardinality of an FD-set in G. In this paper, aiming to provide a particular answer to a problem posed in [Y. Caro, A. Hansberg and M.A. Henning, Fair domination in graphs, Discrete Math. 312 (2012) 2905–2914], we present a new upper bound for the fair domination number of a cactus graph, and characterize all cactus graphs G achieving equality in the upper bound of fd1(G).
Keywords: fair domination, cactus graph, unicyclic graph 
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Hajian, Majid; Rad, Nader Jafari. Fair Domination Number in Cactus Graphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 489-503. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a12/

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