In this paper, we show that the Italian domination number of every lexicographic product graph G ○ H can be expressed in terms five different parameters . These defined under following unified approach, which encompasses definition several well-known and introduces new ones. Let N ( v ) denote open neighbourhood ∈ V , let w = 0 1 …, l a vector nonnegative integers such ≥ We say function f : → {...

A Roman dominating function on a graph G = (V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V ) = ∑ u∈V f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this pape...

A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number of G, γR(G), is the minimum weight of a Roman dominating function on G. In this paper, we...

A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number γR(G) of G is the minimum weight of a Roman dominating function on G. In this paper, we s...

Given an undirected graph G = (V,E) and a weight function w : E → R, the Minimum Dominating Tree problem asks to find a minimum weight sub-tree of G, T = (U,F ), such that every v ∈ V \U is adjacent to at least one vertex in U . The special case when the weight function is uniform is known as the Minimum Connected Dominating Set problem. Given an undirected graph G = (V,E) with some subsets of ...

A two-valued function f defined on the vertices of a graph G = (V,E), f : V → {−1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f(N(v)) ≥ 1, where N(v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f(V ) = ∑ f(v), over all vertices v ∈ V . The total ...

‎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 {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled‎, wher NG(v)  is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎$omega (f):=sum _{vin V(G)}|f(v)|$‎. ‎The 2-rainbow‎‎d...

A three-valued function f defined on the vertices of a graph G = ( V, E), f : V 4 {-I. 0. I }, is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every 1~ t V, ,f(N[o]) > 1, where N[c] consists of I: and every vertex adjacent to 1’. The weight of a minus dominating function is f(V) = c f(u), over all vertices L: t V. The m...