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A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of an RDF f is the value f (V (G)) = u∈V (G) f (u). The Roman domination number of a graph G, denoted by γ R (G), is the minimum weight of a Roman dominating function on G. A graph G is Roman domination stable if the Roman domination number of G remains unchanged under removal of any vertex. In this paper we present upper bounds for the Roman domination number in the class of Roman domination stable graphs, improving bounds posed in [V. Samodivkin, Roman domination in graphs:

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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 f d k (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 f d(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 f d 1 (G).

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 \S. The k-fair total domination number of G, denoted by f td k (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 f td(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.