A subset ‫ܦ‬ of a vertex set ܸሺ‫ܩ‬ሻ of a graph ‫ܩ‬ ൌ ሺܸ, ‫ܧ‬ሻ is called an equitable dominating set if for every vertex ‫ݒ‬ ‫א‬ ܸ െ ‫ܦ‬ there exists a vertex ‫ݑ‬ ‫א‬ ‫ܦ‬ such that ‫ݒݑ‬ ‫א‬ ‫ܧ‬ሺ‫ܩ‬ሻ and |݀݁݃ሺ‫ݑ‬ሻ െ ݀݁݃ሺ‫ݒ‬ሻ| 1, where ݀݁݃ሺ‫ݑ‬ሻ and ݀݁݃ሺ‫ݒ‬ሻ are denoted as the degree of a vertex ‫ݑ‬ and ‫ݒ‬ respectively. The equitable domination number of a graph ߛ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable dominating set of ‫.ܩ‬ An equitable dominating set ‫ܦ‬ is said to be an equitable total dominating set if the induced subgraph ‫ۄܦۃ‬ has no isolated vertices. The equitable total domination number ߛ ௧ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable total dominating set of ‫.ܩ‬ In this paper, we initiate a study on new domination parameter equitable total domination number of a graph, characterization is given for equitable total dominating set is minimal and also discussed Northaus-Gaddum type results

B Basavanagoud

V R Kulli

Vijay V Teli

CiteSeerX

ISSN 0976-5727 (Print) 
ISSN 2319-8133 (Online) 
  Abbr:J.Comp.&Math.Sci. 
2014, Vol.5(2): Pg.235-241 
JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES 
An International Open Free Access, Peer Reviewed Research Journal 
www.compmath-journal.org 
 
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 
Equitable Total Domination in Graphs 
 
B. Basavanagoud1, V. R. Kulli2 and Vijay V. Teli1 
 
1Department of Mathematics, 
Karnatak University, Dharwad, INDIA.  
2Department of Mathematics, 
Gulbarga University, Gulbarga, INDIA. 
 
(Received on: April 16, 2014) 
 
ABSTRACT 
 
A subset  of a vertex set  of a graph   ,  is called an 
equitable dominating set if for every vertex 	 
    there exists 
a vertex  
  such that 	 
  and |  	|  1, 
where  and 	 are denoted as the degree of a vertex  
and 	 respectively. The equitable domination number of a graph 
 of  is the minimum cardinality of an equitable dominating 
set of . An equitable dominating set  is said to be an equitable 
total dominating set if the induced subgraph  has no isolated 
vertices. The equitable total domination number  of  is the 
minimum cardinality of an equitable total dominating set of . In 
this paper, we initiate a study on new domination parameter 
equitable total domination number of a graph, characterization is 
given for equitable total dominating set is minimal and also 
discussed Northaus-Gaddum type results. 
 
2010 Mathematics Subject Classification: 05C69. 
 
Keywords: total domination number, equitable domination 
number, equitable total domination number. 
  
1. INTRODUCTION  
 
The graph   ,  we mean a 
finite, undirected with neither loops nor 
multiple edges. The order and size of  are 
denoted by  and 	 respectively. For graph 
theoretic terminology we refer to Harary3 
and Haynes et al.4 
For any vertex 
  , the open 
neighborhood and closed neighborhood of 
 
are denoted by 
 and 
  
 

 respectively. Degree of a vertex 
 is 
236 B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.5 (2), 235-241 (2014) 
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 
denoted by 
. The maximum 
(minimum) degree of  is denoted by Δ 
(). 
The diameter of a connected graph 
is the maximum distance between two 
vertices in , and is denoted by . 
The length of a shortest cycle is the girth of 
 and is denoted by . For any real 
number , ⌊   denotes the largest integer 
less than or equal to . ⌈   denotes the 
smallest integer greater than or equal to . 
Northaus-Gaddum type result is a (tight) 
lower or upper bound on the sum or product 
of a parameter of a graph and its 
complement. 
A subset  of  is called a 
dominating set of  if every vertex in    
is adjacent to at least one vertex in . The 
minimum cardinality of a minimal 
dominating set is called the domination 
number of  and is denoted by  . 
Various types of domination parameters 
have been defined and studied by several 
authors and more than seventy five models 
of domination parameters are listed in the 
appendix of Haynes et al.4 Cockayne et al.2 
introduced the concept of total domination in 
graphs. A dominating set  of  is called a 
total dominating set if !" has no isolated 
vertices. The minimum cardinality of a total 
dominating set of  is called the total 
domination number of  and is denoted by 
 . 
 
Swaminathan et al.10 introduced the 
concept of equitable domination in graphs, 
by considering the following real world 
problems; In a network nodes with nearly 
equal capacity may interact with each other 
in a better way. In this society persons with 
nearly equal status, tend to be friendly. In an 
industry, employees with nearly equal 
powers form association and move closely. 
Equitability among citizens in terms of 
wealth, health, status etc is the goal of a 
democratic nation. 
In order to study this practical 
concept a graph model is to be created and 
defined as follows: 
A subset  of  is called an 
equitable dominating set if for every vertex 

     there exists a vertex #   such 
that #
   and |#  
| %
1. The minimum cardinality of an equitable 
dominating set of  is called the equitable 
domination number of  and is denoted by 
 . 
In this paper, we use this idea to 
develop the concept of equitable total 
dominating set and equitable total 
domination number of a graph. 
A subset D of V is called an 
equitable total dominating set of G, if D is an 
equitable dominating set and !D" has no 
isolated vertices. The minimum cardinality 
taken over all equitable total dominating sets 
is the equitable total domination number and 
is denoted by γG8. 
 
Figure 1 
 
 The equitable total dominating sets 
of graph  in Figure 1, are:   2, 3, 5, 6,   1, 3, 4, 6,   1, 2, 4 and  2, 4, 5. Therefore,    3. 
 B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.5 (2), 235-241 (2014) 237 
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 
 In this paper, we initiate a study on 
equitable total dominating sets in graphs. 
This parameter is analogous to the total 
domination number which has already been 
studied by Cockayne, Dawes and Hedetniemi 
in2. 
 
2.  MAIN RESULTS 
 
 By the definition of an equitable 
total dominating set, the following result is 
obvious.  
 
Theorem 2.1 
 
 A total dominating set  of  is an 
equitable total dominating set if and only if 
for every vertex 
     there exists a 
vertex #   such that #
   and 
|#  
| % 1.  
Next, we list the exact values of   for 
some standard class of graphs. 
 
Theorem 2.2   
1.  For any path P	 with p 3 2 vertices, 
γP               ;  if  p    0  mod  4 

  1  ;  otherwise .  
2.  For any cycle C	 with p 3 3 vertices, 
γC       ;  if  p    !   2  mod  4 

   ;  p    0  mod  4 .  
3.  For any complete graph K	 with p 3 2 
vertices, γK	  2. 
4. For any wheel W	 with p 3 4 vertices, 
γW  $ 2                 ;  if  p     4, 5 

  1  ;  otherwise .  
5.  For any complete bipartite graph K
,, 
    γK,     2            ;  if  |m  n|      1, 1    m     n m  n  ;  if  |m  n|      2, m, n     2 .  
6.  For any star K,	 with p 3 2 vertices,  
γK,	  p.  
Proof 
(i)   As the degree of any vertex of 7 is 
either 1 or  2. Clearly, any total 
dominating set is equitable. Therefore 
 7   7. 
(ii)  As 8 is a connected 2 - regular graph, 
any total dominating set is equitable. 
Hence  
         8   8. 
(iii) For any complete graph 9 with  3 2, 
any two vertices of 9 forms an 
equitable total dominating set of 9. 
Therefore  9  2. 
(iv) Let : be a wheel, such that : 
#, 
, 
, ; , 
} , where # is the 
center vertex and each 
 ;   1, ; ,   1 is on the cycle. Therefore 
the 	
  3,              
     where 1 %  %   1 and 	# 
  1, hence  3 4. 
We consider the following cases: 
 
Case1. Let  = 4 or 5. 
Then 	#    1 % 4 and  
	
  3, for all , 1 %  %   1. 
Therefore   #  
, 1 %  %   1, is 
an equitable total dominating set of :. 
Hence  :  ||  
                 |#  
|  2. 
 
Case2. Let  3 6. 
In this case 	# 3 5, while  
	  3 for all , where 1      1. 
However   #  
 is a total 
dominating set, but not equitable total 
dominating set. 
 
If   , , ,  ,      1   3  , , ,  , ,      1   3 ! 1 , , ,  , ,      1   3 ! 2 # 
238 B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.5 (2), 235-241 (2014) 
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 
then for any 
    , there exists 
 or 
$   such that, 

 or 

$      
and 
  
  3 or  
  
$  3. Therefore  is 
an equitable  
total dominating set of :. 
Now, 
||    1,      3   2,      3  1     3  2 ." 
Also, when   1  3=, ⌈   =, 
when  
  1  3= > 1 or 3= > 2, ?   = > 1. 
Hence, ||  ?  > 1. 
Therefore  :  ?  > 1. 
 
(v)  Let 9%,& be the complete bipartite graph 
with  vertices in one partition say  and @    
vertices in another partion say . 
Then, '
,#  A,  B  #  
 @,  B  #  
   .C 
If |  @| % 1, then for any vertex #   
and 
   constitute a dominating set 
which is an equitable total dominating set. 
Therefore,  9%,&  2 for |  @| % 1.  
If |  @| 3 2 and || D  > @, where  
is a minimum equitable total dominating set 
of  9%,&, then there exists #    . Let 
#  . 
Therefore '
,#  @. 
  There is a vertex 
  , such that 
 
is adjacent to #   and |'
,
 
'
,#| % 1. Since  is independent, 
 
must belongs to . Therefore '
,
 
. Hence |'
,
  '
,#| 
|  @| 3 2, which is a contradiction. 
Therefore ||   > @. 
Hence  9%,&   > @ for |  @| 3 2 
for all @,  3 2. 
(vi)  If   9,;  3 2, then clearly 
   2. 
 By the definition of equitable total 
dominating set,  should contain all the 
vertices of . Therefore,  9,   ; 
 3 2.  
        From the above results the following 
bound is immediate.  
 
Theorem 2.3.  For any graph  with no 
isolates , 2 %   % . 
 
Corollary 2.1.  For any graph E
9%,&; |  @| 3 2 ;  , @ 3 2 without 
isolated vertices,   % .  
 
Definition 1.  An equitable total dominating 
set is said to be minimal equitable total 
dominating set if no proper subset of D is an 
equitable total dominating set.  
 
 Next theorem gives the 
characterization of the minimal equitable 
total dominating set.  
 
Theorem 2.4.  For any graph G without 
isolated vertices, an equitable total 
dominating set D is minimal if and only if 
for every u  D, one of the following two 
properties holds: 
 
(i) There exists a vertex 
     such that 

 G   #, |#  
| % 1. 
(ii) !  #" contains no isolated vertices.  
  
Proof.  Assume that  is a minimal 
equitable total dominating set and (i) and (ii) 
do not hold. Then for some #  , there 
exists 
     such that |# 

| % 1 and for every 
    , 
either 
 G  E # or |# 
 B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.5 (2), 235-241 (2014) 239 
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 

| 3 2 or both. Therefore !  #" 
contains an isolated vertex, contradiction to 
the minimality of . Therefore (i) and (ii) 
holds. 
 Conversely, if for every vertex 
#  , the statement (i) or (ii) holds and  is 
not minimal. Then there exists #   such 
that   # is an equitable total dominating 
set. Therefore there exists 
    # such 
that 
 equitably dominates #. That is, 

  # and |#  
| % 1. 
Hence # does not satisfy (i). Then # must 
satisfy (ii) and there exists 
     such 
that 
 G   # and |# 

| % 1. And also there exists H   
# such that H is adjacent to 
. Therefore 
H  
 G , |H  
| % 1 
and H E #, a contradiction to 
 G  
#. Hence  is a minimal equitable total 
dominating set.  
 
Proposition 2.1.  For any graph  without 
isolated vertices,   %  .  
 
Proof. Every equitable total dominating set 
is a total dominating set. Thus   % .  
 
Theorem 2.5.  If  is a I regular for 
I 3 1 or (k, k+1) bi-regular for any positive 
integer =, then    .  
 
Proof.  Suppose  is a regular graph. Then 
every vertex of  is of same degree say =. 
Let  be the minimal total dominating set of 
, then ||   . If #    , then  is 
a total dominating set, then there exists 

   and #
  , also # 

  =. Therefore |# 

|  0 % 1. Hence  is an equitable 
total dominating set of , such that   %||   . And also we have   % . Therefore     . 
 Now, suppose  is a =, = > 1 bi-
regular graph. Then the degree of each 
vertex in  is either = or = > 1, where = is a 
positive integer. Let  be a minimal total 
dominating set of , i.e ||    and #    , then there exists 
   such that 
#
   and one of the vertex # or 
 is 
with degree = and other is with degree = >
1. This implies |#  
|  1. 
Therefore  is an equitable total dominating 
set of . Hence   % ||   . But 
also we have that   %  . 
Therefore,     .  
 
In next theorem we give the upper bound for 
  in terms of order and maximum 
degree of .  
 
Theorem 2.6.  For any connected graph 
E 9%,&; |  @| 3 2; , @ 3 2 contains 
no isolated vertices and K D   1, then 
  %   K.  
  
Proof. Let 
 be a vertex with maximum 
degree Δ in  and L    
. If 
L  M, then Δ    1 and     Δ > 1  2, but Δ D   1, 
therefore L E M. Let   L is adjacent to 
N  
. Let 8, ΔO be the vertex set and 
maximum degree of the component of L 
which contains . The component of 8 
has a total equitable dominating set P of 
cardinality at most |8|  ΔO > 1. 
If ΔO  1 then |8|  2 and the set 
, N,  
L  8 is equitable total dominating in  
and   % 3 >   ΔO  1  2 
       Δ. 
If ΔO Q 1, then the set 
, N  P  L 
8 is equitable total dominating in  and 
  % 2 > |8|  ΔO > 1 >   Δ 1  |8|. 
240 B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.5 (2), 235-241 (2014) 
Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 
          Δ > 2  ΔO 
       %   Δ. 
Therefore   %   Δ.  
  
Theorem 2.7.  If E 9%,&; |  @| 3
2; , @ 3 2 be a graph with   2, 
then   %  > 1 and this bound is 
sharp. 
  
Proof.  Let 
   and 
  . 
Since   2, 
 is a dominating 
set in ,   
  
 is an equitable 
total dominating set of  and ||  #  1.  
Hence    %  > 1. 
  We know that  8(  3,  8(  2, and 8(  2, then   8(  8( > 1. 
   In the following theorem we 
establish the upper bound for   in terms 
of order and girth of .  
 
Theorem 2.8.  For any connected graph 
with girth  3 5, and  3 2, 
   %   ?)*+,  > 1.  
  
Proof.  Let  be a connected graph with 
girth  3 5 and  3 2. Now let us 
remove the cycle 8 of shortest length from  
to form O. Suppose an arbitrary vertex 

  O, then 
 has at least two neighbors 
say  and N. If , N  8 and , N 3 3, 
then by replacing the path from  to N on 8 
with the path 
N which reduces the girth of 
, a contradiction. If , N % 2, then 
, N, 
 are on either 8 or 8 in , 
contradiction to the hypothesis that  3
5. Hence no vertex in O has two or more 
neighbors on 8. 
 
Therefore,   %   ?)*+,  > 1.  
Theorem A7. For any graph  without 
isolated vertices,   3 ? -*+,$ 
      Next theorem gives the lower bound 
for   in terms of order and maximum 
degree of .  
 
Theorem 2.9.  For any nontrivial connected 
graph , ? $-*+, > 1 %  .  
 Next theorem gives Nordhaus-
Gaddum type results for  .  
 
Theorem 2.10.  If E 9%,&; |  @| 3
2; , @ 3 2 has  vertices, no isolated 
vertices and K D   1, then 
1.   >   % 2? 
2.   R   % ? 
   Further, equality holds for   8 
and these bounds are sharp.  
 
ACKNOWLEDGEMENT  
 
 This research was supported by 
UGC-SAP DRS-II New Delhi, India: for 
2010-2015.  
 This research was supported by the 
University Grants Commission, New Delhi, 
India. No.F.4-1/2006(BSR)/7-101/2007  
(BSR) dated: 20th June, 2012.  
 
REFERENCES  
 
1. E. J. Cockayne and S. T. Hedetniemi, 
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3. F. Harary, Graph Theory, Addison-
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4. T. W. Haynes, S. T. Hedetniemi and P. 
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in Graphs, Marcel Dekker, Inc., New 
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Equitable total domination in graphs

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Equitable total domination in graphs

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