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A dominating set D in a graph G is called an injective equitable dominating set (Inj-equitable dominating set) if for every , there exists such that u is adjacent to v and . The minimum cardinality of such a dominating set is denoted by and is called the Inj-equitable domination number of G. In this paper, we introduce the injective equitable domination of a graph and study its relation with other domination parameters. The minimal injective equitable dominating set, the injective equitable independence number , and the injective equitable domatic number are defined.

By a graph

A set D of vertices in a graph

The injective domination of graphs has been introduced by A.Alwardi et al. [

A subset D of V is called equitable dominating set of G if every vertex

The importance of injective and equitable domination of graphs motivated us to introduce the injective equitable domination of graphs which mixes the two concepts.

As there are a lot of applications of domination, in particular the injective and equitable domination, we are expecting that our new concept has some applications.

Definition 1 A subset D of

It is easy to see that any Inj-equitable dominating set in a graph G is also a domi- nating set, and then

In the following propostion the Inj-equitable domination number of some standard graphs are determined.

Proposition 1

1) For any complete graph

2) For any path

3) For any cycle

4) For any complete bipartite graph

5) For any wheel graph

Definition 1 motivated us to define the inherent Inj-equitable graph of any graph G as follows:

Definition 2 Let

Theorem 2: For any graph

Proof. Since any Inj-equitable dominating set of

Definition 3 The Inj-equitable neighborhood of

The cardinality of

Definition 4 For any graph G, an edge

Proposition 3 For any graph

Proof. Let G be a graph and let H be the Inj-equitable graph of G. Then

H is the number of Inj-equitable edges in G, then q equals

is equal to

Definition 5 Let

Definition 6 A graph G is called Inj-equitable totally disconnected graph if it has no Inj-equitable edge.

Theorem 4 For any graph G with n vertices,

Proof. It is obviously that

Now, we want to prove that

conversely, suppose that there exists at least one vertex v in G such that

To prove that

Conversely, suppose that G has at least one Inj-equitable edge, say

Proposition 5 If a graph G has no Inj-equitable isolated vertices, then

In the following theorem, we present the graph for which

Theorem 6 Let G be a graph such that any two adjacent vertices contained in a triangle or G is regular triangle-free graph. Then,

Proof. Suppose that G is a regular triangle-free graph and D is a

Let G be a graph such that any two adjacent vertices contains in a triangle. It is clear that for any

Lemma 1 For any two graphs

Proof. Let

That is,

To prove

From 1 and 2, we get

By mathematical induction, we can generalize Lemma 1 as follows:

Proposition 7 Let

Theorem 8 Let G be a graph with

Proof. Let G be a graph with

Conversely, let

Definition 7 An Inj-equitable dominating set D is said to be a minimal Inj-equitable dominating set if no proper subset of D is an Inj-equitable dominating set. A minimal Inj-equitable dominating set D of maximum cardinality is called

The following theorem gives the characterization of the minimal Inj-equitable domi- nating set .

Theorem 9 An Inj-equitable dominating set D is minimal if and only if for every vertex

1) u is not Inj-equitable adjacent to any vertex in D.

2) There exists a vertex

Proof. Suppose that D is minimal Inj-equitable dominating set and suppose that

Conversely, suppose that D is an Inj-equitable dominating set and for every vertex

Theorem 10 A graph G has a unique minimal Inj-equitable dominating set if and only if the set of all Inj-equitable isolated vertices forms an Inj-equitable dominating set.

Proof. Let G has a unique minimal Inj-equitable dominating set D and let

Conversely, let

Theorem 11 If G is a graph has no Inj-equitable isolated vertices, then the com- plement

Proof. Let S be any minimal Inj-equitable dominating set of G and

Theorem 12 For any graph with n vertices

Proof. Let S be a

Thus,

Now,

Therefore,

Hence,

Definition 8 Let

Definition 9 An Inj-equitable independent set S is called maximal if any vertex set properly containing S is not Inj-equitable independent set. The lower Inj-equitable independent number

Theorem 13 Let S be a maximal Inj-equitable independent set. Then S is a minimal Inj-equitable dominating set.

Proof. Let S be a maximal Inj-equitable independent set. Let

Theorem 14 For any graph G,

The maximum order of a partition of a vertex set V of a graph G into dominating sets is called the domatic number of G and is denoted by

Definition 10 An Inj-equitable domatic partition of a graph G is a partition

Example 1 For the graph G given in

Proposition 15

1) For any path

2) For any cycle

3) For any complete graph

4) For any complete bipartite graph

Proposition 16 For any graph G,

Proof. Since any partition of V into Inj-equitable dominating set is also partition of V into dominating set,

In this paper, we introduced the Inj-equitable domination of graphs and some other related parameters like Inj-equitable independent number, uper Inj-equitable domi- nation number and domatic Inj-equitable domination number.

There are many other related parameters for future studies like connected Inj- equitable domination, total Inj-equitable domination, independent Inj-equitable domi- nation, split Inj-equitable domination and clique Inj-equitable domination.

Alkenani, A.N., Alashwali, H. and Muthana, N. (2016) On the Injective Equitable Domination of Graphs. Applied Mathematics, 7, 2132-2139. http://dx.doi.org/10.4236/am.2016.717169