A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G − S is connected. The outer-connected domination number of G, denoted by γ̃c(G), is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate c...

A Roman dominating function (RD-function) on a graph G = ( V , E ) is f : → {0, 1, 2} satisfying the condition that every vertex u for which 0 adjacent to at least one v 2. An in perfect (PRD-function) if with exactly The (perfect) domination number γ R p )) minimum weight of an . We say strongly equals ), denoted by ≡ γR RD-function PRD-function. In this paper we show given it NP-hard decide w...

A Roman dominating function of a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(...

A set D ⊆ V of a graph G = (V,E) is called an outer-connected dominating set of G if for all v ∈ V , |NG[v] ∩ D| ≥ 1, and the induced subgraph of G on V \D is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality of the input graph G. Given a positive integer k and a graph G = (V,E), the Outer-connected Domination Decision ...

Let G = (V,E) be a graph. A set D ⊆ V is a total outer-connected dominating set of G if D is dominating and G[V −D] is connected. The total outer-connected domination number of G, denoted γtc(G), is the smallest cardinality of a total outer-connected dominating set of G. It is known that if T is a tree of order n ≥ 2, then γtc(T ) ≥ 2n 3 . We will provide a constructive characterization for tre...

Let G = (V,E) be a graph. A subset S ⊆ V is a dominating set of G if every vertex not in S is adjacent to a vertex in S. A set D̃ ⊆ V of a graph G = (V,E) is called an outer-connected dominating set for G if (1) D̃ is a dominating set for G, and (2) G[V \ D̃], the induced subgraph of G by V \ D̃, is connected. The minimum size among all outer-connected dominating sets of G is called the outerconnec...

An outer-connected dominating set for an arbitrary graph G is a set D̃ ⊆ V such that D̃ is a dominating set and the induced subgraph G[V \ D̃] be connected. In this paper, we focus on the outerconnected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds f...

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...