Fair domination number in cactus graphs

Majid Hajian | Nader Jafari Rad

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URL :   http://research.shahed.ac.ir/WSR/WebPages/Report/PaperView.aspx?PaperID=85635
Date :  2018/10/14
Publish in :    Discussiones Mathematicae Graph Theory
DOI :  https://doi.org/10.7151/dmgt.2088
Link :  https://doi.org/10.7151/dmgt.2088
Keywords :FAIR, DOMINATION, CACTUS

Abstract :

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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 fair N(v) ∩ Sif = 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).