an equitable domination has interesting application in the contextof social networks. in a network, nodes with nearly equal capacitymay interact with each other in a better way. in the societypersons with nearly equal status, tend to be friendly. in thispaper, we introduce new variant of equitable domination of agraph. basic properties and some interesting results have beenobtained.

Let δ(G), ∆(G) and γ(G) be the minimum degree, maximum degree and domination number of a graph G = (V (G), E(G)), respectively. A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition of G. The maximum number of classes of a domatic partition of G is called the domatic number of G, denoted d(G). It is well known that d(G) ≤ δ(G)+1, d(G)γ(G) ≤ |V (G)| [...

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...

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...

A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition of G. The maximum number of classes of a domatic partition of G is called the domatic number of G. The concept of a domatic number was introduced in [1]. More interesting results on domatically full graphs, domatically critical, domatically cocritical graphs and other domatic numbers can be found ...

We resolve the problem posed as the main open question in [4]: letting δ(G), ∆(G) and D(G) respectively denote the minimum degree, maximum degree, and domatic number (defined below) of an undirected graph G = (V,E), we show that D(G) ≥ (1−o(1))δ(G)/ ln(∆(G)), where the “o(1)” term goes to zero as ∆(G) → ∞. A dominating set of G is any set S ⊆ V such that for all v ∈ V , either v ∈ S or some nei...

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...

Using a dominating set as a coordinator in wireless networks has been proposed in many papers as an energy conservation technique. Since the nodes in a dominating set have the extra burden of coordination, energy resources in such nodes will drain out more quickly than in other nodes. To maximize the lifetime of nodes in the network, it has been proposed that the role of coordinators be rotated...

Wireless sensor networks propound an algorithmic research problems for prolonging life of nodes and network. The domination algorithms can address some of fundamental issues related to lifetime problems in ad hoc and sensor networks. Most of the graph domination problems are NP-complete even with unit-disk-graphs. The investigation of the thesis addresses some of lifetime issues in sensor netwo...

Let G = (V,E) be a simple undirected graph, and k be a positive integer. A k-dominating set of G is a set of vertices S ⊆ V satisfying that every vertex in V \ S is adjacent to at least k vertices in S. A k-domatic partition of G is a partition of V into k-dominating sets. The k-domatic number of G is the maximum number of k-dominating sets contained in a k-domatic partition of G. In this paper...