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Let G = (V;E) be a given graph of order n. A function f: V ! f0; 1; 2g is an independent Roman dominating function (IRDF) on G if for every vertex v 2 V with f(v) = 0 there is a vertex u adjacent to v with f(u) = 2 and fv 2 V: f(v) > 0g is an independent set. The weight of an IRDF f on G is the value f(V) = P v2V f(v). The minimum weight of an IRDF among all IRDFs on G is called the independent Roman domination number of G. In this paper, we give algorithms for computing the independent Roman domination number of G in O(jV j) time when G = (V;E) is a tree, unicyclic graph or proper interval graph. [ABSTRACT FROM AUTHOR]

Let k 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function on a graph G is a function f: V (G) ! f1; 1; 2g such that P u2N(v) f(u) ≥ k for every v 2 V (G), where N(v) is the neighborhood of v, and each vertex u with f(u) = 1 is adjacent to a vertex v with f(v) = 2 or to two vertices w and z with f(w) = f(z) = 1. A set ff1; f2;: ::; fdg of distinct signed total Italian k-dominating functions on G with the property that Pd i=1 fi(v) ≤ k for each v 2 V (G), is called a signed total Italian k-dominating family (of functions) on G. The maximum number of functions in a signed total Italian k-dominating family on G is the signed total Italian k-domatic number of G, denoted by dk stI (G). In this paper we initiate the study of signed total Italian k-domatic numbers in graphs, and we present sharp bounds for dk stI (G). In addition, we determine the signed total Italian k-domatic number of some graphs. [ABSTRACT FROM AUTHOR]