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A double Roman dominating function (DRDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 , 3 } satisfying (i) if f (v) = 0 then there must be at least two neighbors assigned two under f or one neighbor w with f (w) = 3 ; and (ii) if f (v) = 1 then v must be adjacent to a vertex w such that f (w) ≥ 2. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on G if the set of vertices labeled 0 induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number γ tdR oi (G) is the minimum weight of an OITDRDF on G. In this paper, we establish various bounds on γ tdR oi (G). In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results. [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple graph with vertex set V and edge set E. An {outer-independent total 2 -rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2} such that the following conditions hold: (i) for any vertex v with f(v)=∅ we have ⋃u∈NG(v)f(u)={1,2}, (ii) the set of all vertices v∈V(G) with f(v)=∅ is independent and (iii) {v∣f(v)≠∅} has no isolated vertex. The outer-independent total 2 -rainbow domination number of G, denoted by γoitr2(G), is the minimum value of ω(f)=∑v∈V(G)|f(v)| over all such functions f. In this paper, we study the outer-independent total 2 -rainbow domination number of G and classify all graphs with outer-independent total 2 -ainbow domination number belonging to the set {2,3,n}. Among other results, we present some sharp bounds concerning the invariant. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a graph and let f : V (G) → { 0 , 1 , 2 } be a function. A vertex v is protected with respect to f , if f (v) > 0 or f (v) = 0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex in V is protected, and (ii) each u ∈ V with positive weight has a neighbor v ∈ V with f (v) = 0 such that the function f u v : V → { 0 , 1 , 2 } , defined by f u v (v) = 1 , f u v (u) = f (u) − 1 and f u v (x) = f (x) for x ∈ V ∖ { v , u } , has no unprotected vertex. The weight of f is ω (f) = Σ v ∈ V f (v). The co-Roman domination number of a graph G , denoted by γ c r (G) , is the minimum weight of a co-Roman dominating function on G. In this paper, we present some new sharp bounds on γ c r (G). Some of our results improve the previous bounds. [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V (G). A signed Roman k-dominating function (SRkDF) on a graph G is a function f : V (G) → {−1, 1, 2} such that (i) every vertex v with f (v) = −1 is adjacent to at least one vertex u with f (u) = 2, (ii) Σu ϵN[v]f (u) ≥ k holds for any vertex v. The weight of a SRkDF f is ΣuϵV (G)f (u), and the minimum weight of a SRkDF is the signed Roman k-domination number (G) of G. In this paper, we investigate the (G) signed Roman k-domination number of graphs, and we establish some bounds on (G). In the case that T is a tree, we present lower and upper bounds on (T) for k ∈ {3, 4} and classify all extremal trees. [ABSTRACT FROM AUTHOR]

A nonnegative signed edge dominating function of a graph G = (V,E) is a function f : E → {-1, 1} such that Σe'∊N[e] f(e') ≥ 0 for each e ∊ E, where N[e] is the closed neighborhood of e. The weight of a nonnegative signed edge dominating function f is !(f) = Σ e∊E f(e). The nonnegative signed edge domination numberγ'ns(G) of G is the minimum weight of a nonnegative signed edge dominating function of G. In this paper, we prove that for every tree T of order n ≥ 3, 1 - n/3 ≤γ'ns(T) ≤[n-1/3] . Also we present some sharp bounds for the nonnegative signed edge domination number. In addition, we determine the nonnegative signed edge domination number for the complete graph, and the complete bipartite graph Kn,n. [ABSTRACT FROM AUTHOR]