Geodesic
A Note on the Fair Domination Number in Outerplanar Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1085-1093

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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, we present a new sharp upper bound for the fair domination number of an outerplanar graph.
Keywords: fair domination, outerplanar graph, unicyclic graph 
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Hajian, Majid; Rad, Nader Jafari. A Note on the Fair Domination Number in Outerplanar Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1085-1093. http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a9/