Some Results on Equi Independent Equitable Dominating Sets in Graph

A subset D of ( ) V G is called an equitable dominating set if for every ( ) v V G D   , there exists a vertex u D  such that ( ) uv E G  and | ( ) ( ) | 1 deg u deg v   . A subset D of ( ) V G is called an equitable independent set if for any u D  , ( ) e v N u  for all { } v D u   where, ( ) { ( ) / ( ), e N u v V G v N u    | ( ) ( ) | 1} deg u deg v   . An equitable dominating set D is said to be an equi independent equitable dominating set if it is also an equitable independent set. The minimum cardinality of an equi independent equitable dominating set is called equi independent equitable domination number which is denoted by e i . We investigated an equi independent equitable domination number for some special graphs.

 .An equitable dominating set D is said to be an equi independent equitable dominating set if it is also an equitable independent set.The minimum cardinality of an equi independent equitable dominating set is called equi independent equitable domination number which is denoted by e i .We

Introduction
Throughout this work, the term graph we mean finite, connected, undirected and simple graph G with vertex set () VGand edge set () EG .For any undefined term we rely upon West [9] and Haynes et al. [5].For every vertex ()  is adjacent to at least one vertex in D .The domination number () G  is the minimum cardinality of a dominating set of G .A subset D of () VG is an independent set if no two vertices in D are adjacent.A dominating set D which is also an independent set is called an independent dominating set.The independent domination number () iG is the minimum cardinality of an independent dominating set.The concept of an independent domination was formalized by Berge [2] and Ore [6] while the definition of an independent domination number and the notation () iG were introduced by Cockayne and Hedetniemi [3].A survey on the concept of an independent domination can be found in Goddard and Henning [4] while applications of dominating sets in computer network is well studied by Basavanagoud and Hosamani [1].
A subset D of () VGis called an equitable dominating set if for every The minimum cardinality of such an equitable dominating set is equitable domination number of G which is denoted by e  .An equitable dominating set D is said to be a minimal equitable dominating set if no proper subset of D is an equitable dominating set.Swaminathan and Dharmalingam [7] have derived following necessary and sufficient condition for minimal equitable dominating set.
Theorem 1.1:An equitable dominating set D is minimal if and only if for every vertex uD  one of the following holds.
(i) Either ()  [7] have also introduced the concept of equitable independent set.According to them a subset D of () VGis called an equitable independent set if for any uD  , () . The maximum cardinality of an equitable independent set is denoted by e  .
Remark 1.5: Every independent set is an equitable independent set.
Remark 1.6: [7] Let D be a maximal equitable independent set.Then D is a minimal equitable dominating set.Motivated by the concept of equitable dominating set and equitable independent set a new concept was conceived by Swaminathan and Dharamlingam [7] while it was formalized and named as equi independent equitable dominating set by Vaidya and Kothari [8].Definition 1.7:An equitable dominating set D is said to be equi independent equitable dominating set if it is also equitable independent set.The minimum cardinality of an equi independent equitable dominating is called equi independent equitable domination number which denoted by e i .

Definition 2.1:
The corona G 1 ʘG 2 of two graphs 1 G and 2 G is defined to be the graph obtained by taking one copy of 1 G of order is an equi independent equitable dominating set with minimum cardinality.Hence, i e (P 2 ʘK 1 ) = γ e (P 2 ʘ K 1 ) = 2.

Case 3:
4 n  In P 4 ʘK 1 , vertices 23 , uu are equitable isolates and 14 , uu are pendant vertices.Then is an equi independent equitable dominating set of P 4 ʘK 1 with minimum cardinality.Hence, i e (P n ʘK 1 ) = γ e (P 4 ʘK 1 ) = 4 Case 4: are equitable isolates of P n ʘK 1 .Therefore the set must be a subset of every equitable dominating set.While vertex 1 u is equitably adjacent to only 1 v and vertex n u is equitably adjacent to only n v .Therefore one of the pair from 1 { , } n vv or 1 { , } n uu must belongs to every equitable dominating set which implies that i e (P n ʘK 1 ) ≥ i e (P n-4 ) + n Now depending upon the number of vertices of path n P , consider the following subsets.
For 0( mod 3) n  , where 02 3 where 02  3 Therefore N e [D] = V(P n ʘK 1 ) and D is an equitable dominating set of P n ʘK 1 .Also D is an equitable independent set as pendant vertices Proof: Let 12 , , , n v v v  be the vertices of degree 3 and 12 , , , n u u u  vertices of degree1 of crown C n ʘK 1 .Observe that vertices 12 , , , n u u u  are equitable isolates of C n ʘK 1 .This implies that they must belong to every equitable dominating set which implies that i e (C n ʘK 1 ) ≥ γ e (C n ʘK 1 ) ≥ γ e (C n ) + n).
Let S be the e  .We claim that D is an equi independent equitable dominating set of C n ʘK 1 .Observe that D is an equitable dominating set of C n ʘK 1 with minimum cardinality as all the equitable isolates belongs to D and remaining all the vertices are dominated by set S .Also D is an equitable independent set as all 12 , , , n u u u  are equitable isolates and vertices of set S are not equitably adjacent to any other vertex of set D .Hence, i e (C n ʘK 1 ) = γ e (C n ʘK 1 ) = γ e (C n ) + n).
and vertices 1 2 3 ,, v u u are non adjacent vertices.Therefore D is an equi independent equitable dominating set of 4 G . Hence, e iG  .

Case 3: 5 n 
In this case apex vertex v is an equitable isolates and set ( )  is an equi independent equitable dominating set with minimum cardinality.Hence, Definition 2.10: The fan n f is graph on 1 n  vertices obtained by joining all the vertices of n P to a new vertex called apex vertex.
Theorem 2.11: Proof: Let v be a apex vertex and 12 , , , n v v v  be the rim vertices of fan n f .
Case 1: equitably dominate all the vertices and D is an equitable independent set.
is an equitable dominating set as well as it is an independent set of n F .Hence, D is an equi independent equitable dominating set of n F and ( ) ( ) 1

Concluding Remarks
Some fundamental results on the concept of equi independent equitable domination number are established by the authors but for the sake of brevity they are not reported here.Here we investigated equi independent equitable domination number for some special graphs.To establish the bounds in terms of various graph theoretic parameters in the context of equi independent equitable domination number is an open area of research.
v  .A subset D of () VGis called an equitable independent set if for any uD  ,

Definition 2 . 4 :Theorem 2 . 5 :
adjacent to any other vertex of D and remaining vertices form   set of n P .Therefore D is an equi independent equitable dominating set of P n ʘK 1 i e (P n ʘK 1 ) = n + i e (P n-4 ).The crown C n ʘK 1 is obtained by joining pendant edge to each vertex of cycle n C .i e (C n ʘK 1 ) = γ e (C n ʘK 1 ) = γ e (C n ) + n).

4 v
are non adjacent to each other.While

degree of graph G respectively. Remark 1.2: ( )
an equitable isolate and D is any equitable dominating set then vD  .Obviously isolated vertices are equitable isolates. .

Remark 1.3: In regular graphs and  
 .Therefore D is an equitable dominating set with minimum cardinality.Also D is an equitable independent set as vertices of D are non adjacent to each other.Therefore D is an equi independent equitable dominating set with || Dn  .Hence, e iG  .
The friendship graph n F is a one-point union of n copies of cycles 3 C .
e if .