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