In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

Let G be an intuitionistic fuzzy graph. Let u and v be two vertices of G. A subset D of V is called a fuzzy equitable dominating set if every v ∈ V − Dthere exist a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| = 1 where deg(u) denotes the degree of vertex u and deg(v) denotes the degree of vertex v and μ2(vi, vj ) = μ1(vi) ∧ μ1(vj ), γ2(vi, vj ) = γ1(vi) ∨ γ1(vj ). The minimum cardina...

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 . ...

In this paper, the concept of incidence domination number of graphs  is introduced and the incidence dominating set and  the incidence domination number  of some particular graphs such as  paths, cycles, wheels, complete graphs and stars are studied.

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...

Let G be an IFG. Then V D  is said to bae a strong (weak) dominating set if every D V v   is strongly (weakly) dominated by some vertex in D. We denote the strong (weak) intuitionistic fuzzy dominating set by sid-set (wid-set). The minimum vertex cardinality over all the sid-set (wid-set) is called the strong (weak) dominating number of an IFG and is denoted by )] ( [ ) ( G G wid sid   In ...

Let G be a fuzzy graph. A subset D of V is said to be Fuzzy dominating set if every vertex u ∈ V (G) there exists a vertex v ∈ V −D such that uv ∈ E(G) and μ(uv) 6 σ(u) ∧ σ(v). The minimum Cardinality of fuzzy dominating set is denoted by γf . A graph G is said to be fuzzy excellent if every vertex of G belongs to γf -sets of G. In this paper, we give a construction to imbedded non-excellent fu...