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) = 0 we have ,u ϵ,NG (v) d(u)={1, 2} (ii) the set of all vertices v ϵ,V (G) with f(v) = 0 is independent and (iii) {v|f(v) ≠, 0} has no isolated vertex. The outer-independent total 2-rainbow domination number of G, denoted by ɤ, oitr2(G), is the minimum value of ω, (f) = ∑, v ϵ,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-rainbow domination number belonging to the set {2,3,n}. Among other results, we present some sharp bounds concerning the invariant.

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Let G = (V; E;  ) be a  nite signed graph. A function f: V! f1; 0; 1g is a minus dominating function (MDF) of G if f(u) + P v2N(u)  (uv)f(v)  1 for all u 2 V. In this paper we characterize signed paths and cycles admitting an MDF.

Let G be a graph. A 2-rainbow dominating function (or 2-RDF) of G is a function f from V (G) to the set of all subsets of the set f1; 2g such that for a vertex v 2 V (G) with f(v) =; , the condition S u2NG(v) f(u) = f1; 2g is ful lled, where NG(v) is the open neighborhood of v. The weight of 2-RDF f of G is the value! (f): = P v2V (G) jf(v)j. The 2-rainbow domination number of G, denoted by r2(G), is the minimum weight of a 2-RDF of G. A 2-RDF f is called an outer independent 2-rainbow dominating function (or OI2-RDF) of G if the set of all v 2 V (G) with f(v) =; is an independent set. The outer independent 2-rainbow domination number oir2(G) is the minimum weight of an OI2-RDF of G. In this paper, we obtain the outer independent 2-rainbow domination number of Pm  Pn and Pm  Cn. Also we determine the value of oir2(Cm2Cn) when m or n is even.

A 2-rainbow dominating function on a graph G is a function g that assigns to each vertex a set of colors chosen from the subsets of {1, 2} so that for each vertex with g(v) = 0 we have ,uϵ, N(υ,) g(u) = {1, 2}. The weight of a 2-rainbow dominating function g is the value ω,(g) = ∑,υ, ϵ, v(G) |f(υ, )|. A 2-rainbow dominating function g is an independent 2-rainbow dominating function if no pair of vertices assigned nonempty sets are adjacent. The 2-rainbow domination number ɤ, r2(G) (respectively, the inde-pendent 2-rainbow domination number ir2(G)) is the minimum weight of a 2-rainbow dominating function (respectively, independent 2-rainbow dominating function) on G. We prove that for any tree T of order n ≥, 3, with l leaves and s support vertices, ir2(T) ≤, (14n + ʆ,+ s)=20, thus improving the bound given in [Independent 2-rainbow domination in trees, Asian-Eur. J. Math. 8 (2015) 1550035] under certain conditions.