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Abstract: A Roman dominating function on a graph is a labeling such that every vertex with label 0 has a neighbor with label 2. A set of Roman dominating functions on with the property that for each is called a Roman dominating family (of functions) on . The maximum number of functions in a Roman dominating family on is the Roman domatic number of , denoted by . In this work we initiate the study of the Roman domatic number in graphs and we present some sharp bounds for . In addition, we determine the Roman domatic number of some graphs. [Copyright &y& Elsevier]

Abstract: For a positive integer , a -rainbow dominating function of a graph is a function from the vertex set to the set of all subsets of the set such that for any vertex with the condition is fulfilled, where is the neighborhood of . The -rainbow domination is the same as the ordinary domination. A set of -rainbow dominating functions on with the property that for each is called a -rainbow dominating family (of functions) on . The maximum number of functions in a -rainbow dominating family on is the -rainbow domatic number of , denoted by . Note that is the classical domatic number . If is a graph of order and is the complement of , then we prove in this note for the Nordhaus–Gaddum inequality This improves the Nordhaus–Gaddum bound given by Sheikholeslami and Volkmann recently. [Copyright &y& Elsevier]

Abstract: A function is a Roman dominating function if every vertex for which is adjacent to at least one vertex for which . A function with the ordered partition of , where for , is a unique response Roman function if implies and implies that . A function is a unique response Roman dominating function if it is a unique response Roman function and a Roman dominating function. The unique response Roman domination number of , denoted by , is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination number of graphs and present bounds for this parameter. [Copyright &y& Elsevier]

Abstract: Let and let be a graph. According to Dunbar et al. [-Domination, Discrete Math. 211 (2000) 11–26], a set is an -dominating set of G if for all . Similarly, we define a set to be an -independent set of G if for all . The -domination number of G is the minimum cardinality of an -dominating set of G and the -independent -domination number of G is the minimum cardinality of an -dominating set of G that is also -independent. A graph G is -domination perfect if for all induced subgraphs H of G. We characterize the -domination perfect trees in terms of their minimally forbidden induced subtrees. For there is exactly one such tree whereas for there are infinitely many. [Copyright &y& Elsevier]

Abstract: Let be a simple graph without isolated vertices with vertex set and edge set . A function is said to be a signed star dominating function on if for every vertex of , where . A set of signed star dominating functions on with the property that for each , is called a signed star dominating family (of functions) on . The maximum number of functions in a signed star dominating family on is the signed star domatic number of , denoted by . In this paper we study the properties of the signed star domatic number . In particular, we determine the signed domatic number of some classes of graphs. [Copyright &y& Elsevier]