The independent domatic queen number of a graph Qn is the maximum pairwise disjoint minimum dominating sets Pn and it denoted by id(Qn) while Id(Qn). We discuss about in this paper (maximum independent) on n × chess board.

Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V \ S is adjacent to a vertex in S as well as to another vertex in V \S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ r (G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper boun...

For any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$‎, ‎we define a‎ ‎function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating‎ ‎function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least‎ ‎$k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$‎. ‎The minimum weight of a Roman $k$-tuple dominatin...

A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. A dominating set is connected if the subgraph induced by its vertices is connected. The connected domatic partition problem asks for a partition of the nodes into connected dominating sets. The connected domatic number of a graph is the size of a largest connected domatic partition and it...

A subset D of the vertex set V (G) of a graph G is called dominating in G, if each vertex of G either is in D, or is adjacent to a vertex of D. If moreover the subgraph 〈D〉 of G induced by D is regular of degree 1, then D is called an induced-paired dominating set in G. A partition of V (G), each of whose classes is an induced-paired dominating set in G, is called an induced-paired domatic part...