Fair Domination Number in Cactus Graphs

For ≥ 1, a -fair dominating set (or just FD-set) in a graph is a dominating set such that | ) ∩ | = for every vertex ∈ \ . The -fair domination number of , denoted by ), is the minimum cardinality of a FD-set. A fair dominating set, abbreviated FD-set, is a FD-set for some integer ≥ 1. The fair domination number, denoted by ), of that is not the empty graph, is the minimum cardinality of an FD-set in . In this paper, aiming to provide a particular answer to a problem posed in [Y. Caro, A. Hansberg and M.A. Henning, , 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 achieving equality in the upper bound of ).