A Roman dominating function (RDF) on a graph G = (V; E) is a function f: V! f0; 1; 2g such that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. An RDF f is called an outer independent Roman dominating function (OIRDF) if the set of vertices assigned a 0 under f is an independent set. The weight of an OIRDF is the sum of its function values over all vertices, and the outer independent Roman domination number oiR(G) is the minimum weight of an OIRDF on G. . .

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.

An outer-independent double Italian dominating function (OIDIDF) on a graph G with vertex set V (G) is a function f: V (G)! f0; 1; 2; 3g such that if f(v) 2 f0; 1g for a vertex v 2 V (G) then P u2N[v] f(u)  3, and the set fu 2 V (G)jf(u) = 0g is independent. The weight of an OIDIDF f is the value w(f) = P v2V (G) f(v). The minimum weight of an OIDIDF on a graph G is called the outer-independent double Italian domination number oidI (G) of G. We present sharp lower bounds for the outer-independent double Italian domination number of a tree in terms of diameter, vertex covering number and the order of the tree.

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.

For a given simple graph G = (V; E), a set S  V is an outer-weakly convex dominating set if every vertex in V nS is adjacent to some vertex in S and V nS is a weakly convex set. The outer-weakly convex domination number of a graph G, denoted by e wcon(G), is the minimum cardinality of an outer-weakly convex dominating set of G. In this paper, we initiate the study of outer-weakly convex domination as a new variant of graph domination and we show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on e wcon(G) and, for some particular families of graphs, we obtain closed formula.

In this paper, we initiate the study of total outer-convex domination as a new variant of graph domination and we show the close relationship that exists between this novel parameter and other domination parameters of a graph such as total domination, convex domination, and outer-convex domination. Furthermore, we obtain general bounds of total outer-convex domination number and, for some particular families of graphs, we obtain closed formulas.

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.

Please click on PDF to view the abstract.