Let $A$ and $B$ be two disjoint subsets of the vertex set $V$ of a graph $G$. The set $A$ is said to dominate $B$, denoted by $A rightarrow B$, if for every vertex $u in B$ there exists a vertex $v in A$ such that $uv in E(G)$. For any graph $G$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_p}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i rightarrow V_j$ or $V_j rightarrow...

A dominating set of a graph G =( P’, E) is a subset D of Vsuch that every vertex not in D is adjacent to some vertex in D. The domatic number d(G) of G is the maximum positive integer k such that V can be partitioned into k pairwise disjoint dominating sets. The purpose of this paper is to study the domatic numbers of graphs that are obtained from small graphs by performing graph operations, su...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {−1, 1} is said to be a signed star k-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that ∑d i=1 fi(e)...

for every positive integer k, a set s of vertices in a graph g = (v;e) is a k- tuple dominating set of g if every vertex of v -s is adjacent to at least k vertices and every vertex of s is adjacent to at least k - 1 vertices in s. the minimum cardinality of a k-tuple dominating set of g is the k-tuple domination number of g. when k = 1, a k-tuple domination number is the well-studied domination...

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1, f2, ....

An edge dominating set in a graph G is a set of edges D such that every edge not in D is adjacent to an edge of D. An edge domatic partition of a graph C=(V, E) is a collection of pairwise-disjoint edge dominating sets of G whose union is E. The maximum size of an edge domatic partition of G is called the edge domatic number. In this paper, we study the edge domatic number of the complete parti...