# MINIMAL DOMINATING FUNCTIONS OF CORONA PRODUCT GRAPH OF A CYCLE WITH A COMPL

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME
TECHNOLOGY (IJCET)

ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)                                                       IJCET
Volume 4, Issue 4, July-August (2013), pp. 248-256
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MINIMAL DOMINATING FUNCTIONS OF CORONA PRODUCT GRAPH
OF A CYCLE WITH A COMPLETE GRAPH

1                      2
M.Siva Parvathi,   B.Maheswari
1
Dept. of Mathematics, K.R.K. Govt. Degree College, Addanki-523201, Andhra Pradesh, India
2
Dept. of Applied Mathematics, S.P.Women’s University, Tirupat-517502, Andhra Pradesh, India

ABSTRACT

‘Domination in graphs’ is the fastest growing area in Graph Theory that has emerged rapidly
in the last three decades. An introduction and an extensive overview on domination in graphs and
related topics is surveyed and detailed in the two books by Haynes et al. [ 1, 2]. Recently, dominating
functions in domination theory have received much attention. Frucht and Harary [3] introduced a
new product on two graphs G1 and G2, called corona product denoted by G1ÍG2. In this paper we
present some results on minimal dominating functions of corona product graph             Í     .

Key Words : Corona Product, Cycle, Complete graph, Dominating Function.

Subject Classification:

1. INTRODUCTION

Domination Theory is an important branch of Graph Theory that has many applications in
Engineering, Communication Networks and many others. Allan, R.B. and Laskar, R.[4], Cockayne,
E.J. and Hedetniemi, S.T. [5] have studied various domination parameters of graphs. An introduction
and an extensive overview on domination in graphs and related topics is surveyed and detailed in the
two books by Haynes et al. [ 1, 2].
Recently, dominating functions in domination theory have received much attention. A purely
graph – theoretic motivation is given by the fact that the dominating function problem can be seen, in
a clear sense, as a proper generalization of the classical domination problem. Jeelani Begum, S. [6]
has studied some dominating functions of Quadratic Residue Cayley graphs.
Product of graphs occur naturally in discrete mathematics as tools in combinatorial
constructions. They give rise to important classes of graphs and deep structural problems. There are
four main products that have been studied in the literature: the Cartesian product, the strong product,
the direct product and the Lexicographic product of finite and infinite graphs.

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME

Frucht and Harary [3] introduced a new product on two graphs G1 and G2, called corona
product denoted by G1ÍG2. The object is to construct a new and simple operation on two graphs G1
and G2 called their corona, with the property that the group of the new graph is in general isomorphic
with the wreath product of the groups of G1 and of G2.
In this paper we present some basic properties of corona product graph               Í      and
some results on minimal dominating functions are obtained.

2. CORONA PRODUCT OF                AND

The corona product of a cycle         with a complete graph      is a graph obtained by taking
one copy of a n – vertex graph       and n copies of    and then joining the vertex of        to every
vertex of    copy of      and it is denoted by
We present some properties of the corona product graph                   without proofs and the
proofs can be found in Siva Parvathi, M. [7].

Theorem 2.1: The graph                  is a connected graph.

Theorem 2.2: The degree of a vertex v in G                  is given by

Theorem 2.3: The number of vertices and edges in                       is given respectively by
1.                     ,
2.                            .

Theorem 2.4: The graph                  is non - hamiltonian.

Theorem 2.5: The graph                  is eulerian if      is even.

Theorem 2.6: The graph                  is not bipartite.

3. DOMINATING SETS AND DOMINATING FUNCTIONS

In this section we prove some results on Minimal Dominating Functions. First let us recall
some definitions.
Definition: Let           be a graph. A subset of    is said to be a dominating set (DS) of if
every vertex in – is adjacent to some vertex in .
A dominating set is called a minimal dominating set (MDS) if no proper subset of is a
dominating set of .
Definition: The domination number of is the minimum cardinality taken over all minimal
dominating sets in and is denoted by γ       .
Definition: Let          be a graph. A function             is called a dominating function (DF)
of if                ∑ f (u ) ≥ 1,
u∈N [v ]
∈

Here       is the closed neighbourhood set of
Definition: Let and be functions from to [0,1]. We define                           if      ≤     for all
∈ , with strict inequality for atleast one vertex

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME

A dominating function of is called a minimal dominating function (MDF) if for all
is not a dominating function.

Theorem 3.1: The domination number of                        is n. i.e.           .
Proof: Let denote a dominating set of .
Case 1: Suppose contains the vertices of            in .
By the definition of , each vertex in          is adjacent to all vertices of ith copy of    . That is the
vertices in      dominate the vertices in all copies of        respectively. Therefore the vertices of
dominate all vertices of
Thus becomes a DS of . This set is also minimal, because, if we delete one vertex say              from ,
then the vertices in the kth copy of       are not dominated by any vertex in .
Case 2: Suppose contains any vertex of each copy of             in .
That is            . Obviously every vertex in       dominates every other vertex in      and also a single
vertex of      to which it is connected. Therefore the vertices in dominate all vertices of . Further
this set is also minimal.
Hence in either case we get              .

Theorem 3.2: Let be a MDS of                           Then a function     defined by
1, if v ∈ D,

0, otherwise.
becomes a MDF of
Proof: Consider the graph                     with vertex set V.
Let be a MDS of G. Then by Theorem 3.1, contains either all the vertices of Cn or any vertex of
each copy of Km in G. For definiteness let contain the vertices of .
Case 1: Let         be such that                     in .
Then       contains m vertices of          and three vertices of   in .
So ∑ f (u ) = 1 + 1 + 1 + 0 + ....... + 0 = 3.
14 4 2 3
u∈N [v ]                        m −times
Case 2: Let          be such that                   in .
Then        contains m vertices of             and one vertex of     in .
So ∑ f (u ) = 1 + 0 + ....... + 0 = 1 .
14 42 3
u∈N [v ]            m − times

Therefore for all possibilities, we get          ∑ f (u ) ≥ 1,
u∈N [v ]
∀   ∈

This implies that is a DF.
Now we check for the minimality of
Define                  by
r, if v = v k ∈ D,

1, if v ∈ D - {v k },
0, otherwise.

where
Since strict inequality holds at the vertex v k ∈ D, it follows that
Now the following cases arise.
Case 1: Let          be such that                 in .
Sub case 1: Let

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
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Then      ∑ g (u ) = r + 1 + 1 + 0 +4243 = r + 2 > 1.
u∈N [v ]
1 ....... + 0
m−times
Sub case 2: Let
Then ∑ g (u ) = 1 + 1 + 1 + 0 + ....... + 0 = 3.
14 4 2 3
u∈N [v ]                              m − times
Case 2: Let        be such that                                        in .
Sub case 1: Let
Then ∑ g (u ) = r + 0 + ....... + 0 = r < 1.
14 4 2 3
u∈N [v ]                  m −times
Sub case 2: Let
Then ∑ g (u ) = 1 + 0 + ....... + 0 = 1.
14 4 2 3
u∈N [v ]                  m − times

This implies that        ∑ g (u ) < 1,
u∈N [v ]
for some           ∈ .

So is not a DF.
Since is taken arbitrarily, it follows that there exists no                                   such that       is a DF.
Thus is a MDF.

Theorem 3.3: A function                                → [ 0, 1 ] defined by                              ∀     ∈    is a DF of
if         . It is a MDF if
Proof: Let V be the vertex set of                       and be a function defined as in the hypothesis.
Case 1: Suppose
Sub case 1: Let            be such that                    in .
Then           contains vertices of           and three vertices of  in .
1       1                 1     m+3
So ∑ f (u ) =          +        + ....... +      =        > 1, since
u∈N [v ]       q +1 q +1                 q +1 q +1
14444 4444   2              3
(m + 3 )− times
Sub case 2: Let                be such that             in .
1    1                1   m +1
So    ∑
u∈N [v ]
f (u ) =     +
q +1 q +1
+ ....... +     =
q +1 q +1
> 1, since
14444244443
(m +1)−times
Therefore for all possibilities, we get                     ∑ f (u ) > 1,
u∈N [v ]
∀   ∈

This implies that is a DF.
Now we check for the minimality of
Define                  by
 r, if v = v k ∈ V ,

 1
 q + 1 , otherwise.

where                   .
Since strict inequality holds at a vertex                             of V, it follows that
Case (i): Let         be such that                                           in G.
Sub case 1: Let

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME

1              1                           1
Then        ∑[ ] g (u ) = r + q + 1 + q + 1 + ....... + q + 1
u∈N v
14444244443
( m+ 2 )−times
1   m+2 m+3
<       +     =      >1 ,
q +1 q +1   q +1
Sub case 2: Let
1              1                        1            m+3
Then        ∑[ ] g (u ) = q + 1 + q + 1 + ....... + q + 1 =
u∈N v                                                                q +1
> 1,
14444244443
( m +3 )−times
Case (ii): Let             be such that                 in G.
Then we can see similarly as above that for the possibilities                                   we have
1       1                 1       1       m m +1
∑[v ] g (u ) = r + q + 1 + q + 1 + ....... + q + 1 < q + 1 + q + 1 = q + 1 > 1,
u∈N
14444 4444   2              3
m −times
and for                     , we have
1       1                 1     m +1
∑[v ] g (u ) = q + 1 + q + 1 + ....... + q + 1 = q + 1 > 1,
u∈N
14444 4444   2              3
(m +1)−times
Hence it follows that           ∑ g (u ) ≥ 1,
u∈N [v ]
∀          ∈

Thus is a DF.
This implies that is not a MDF.
Case 2: Suppose
Substituting q = m in Case 1 we can see that for                 , we have
1       1                 1     m+3 m+3              2
∑[v ] f (u ) = q + 1 + q + 1 + ....... + q + 1 = q + 1 = m + 1 = 1 + m + 1 > 1.
u∈N
14444244443
(m + 3 )−times
and for                , we have
1      1                 1   m +1 m +1
∑
u∈N [v ]
f (u ) =       +
q +1 q +1
+ ....... +     =     =
q +1 q +1 m +1
= 1.
14444244443
(m +1)−times
Therefore for all possibilities, we get                          ∑ f (u ) ≥ 1, ∀
u∈N [v ]
∈

This implies that is a DF.
Now we check for the minimality of .
Define                      by
 r, if v = v k ∈ V ,

 1
 m + 1 , otherwise.

where                     .
Since strict inequality holds at a vertex          of V, it follows that
Then we can see as in case (i), that for                and              , we have
1      1                1          1      m+2 m+3
∑[v ] g (u ) = r + m + 1 + m + 1 + ....... + m + 1 < m + 1 + m + 1 = m + 1 > 1
u∈ N                           4
14444 24444 3              4
( m + 2 )−times

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME

and for              and                , we have
1       1                  1     m+3
∑[v ] g (u ) = m + 1 + m + 1 + ....... + m + 1 = m + 1 > 1.
u∈ N                      4
14444 24444 3               4
( m + 3 )−times
Similarly we can show as in case (ii) that for               and                               we have
1       1                 1       1       m
∑[v ] g (u ) = r + m + 1 + m + 1 + ....... + m + 1 < m + 1 + m + 1 = 1.
u∈N                           4
14444 244444                3
m − times

and for                     and       , we have
1       1                 1      m +1
∑[v ] g (u ) = m + 1 + m + 1 + ....... + m + 1 = m + 1 = 1.
u∈ N                      4
14444 24444 3              4
(m +1)− times
This implies that             ∑ g (u ) < 1,
u∈N [v ]
for some v ∈ V.

So is not a DF.
Since is defined arbitrarily, it follows that there exists no                              such that     is a DF.
Thus is a MDF.

Theorem 3.4: A function f : V → [ 0, 1 ] defined by                                            ∀   ∈      where p = min (m,
n) and       q = max ( m, n ) is a DF if                                    . Otherwise it is not a DF. Also it becomes MDF
if                  .
Proof: Let                              be the given graph with vertex set .
Let      →                        be defined by             ∀ ∈ , where p = min(m, n) and                      q=max ( m, n).
Clearly
Case 1: Let                      be such that             in .
p p            p        p
Then      ∑
u∈N [v ]
f (u ) = + + ....... + = (m + 3) .
q q            q        q
4
144 244 3     4
( m +3 )−times
Case 2: Let                      be such that          in G.
p p            p         p
Then      ∑
u∈N [v ]
f (u ) = + + ....... + = (m + 1) .
q q            q         q
4
144 244 3     4
(m +1)−times
From the above two cases, we observe that                             is a DF if           .
Otherwise is not a DF.
Case 3: Suppose   >    .
Clearly     is a DF.
Now we check for the minimality of                             .
Define                by
r, if v = v k ∈ V ,

p
 q , otherwise.

where                        .

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Since strict inequality holds at a vertex                     of V, it follows that
Case (i): Let         be such that                                   in .
Sub case 1: Let             .
p p              p
Then ∑ g (u ) = r + + + ....... +
u∈N [v ]         q q              q
4
144 244 3      4
( m+ 2 )−times
p             p        p
<  + (m + 2 ) = (m + 3) > 1,                                >        .
q             q        q
Sub case 2: Let v k ∉ N [v ] .
p p            p        p
Then ∑ g (u ) = + + ....... + = (m + 3) > 1.
u∈N [v ]     q q             q        q
4
144 244 3      4
( m +3 )−times
Case (ii): Let      be such that                                  in .
Sub case 1: Let          .
p p            p
Then ∑ g (u ) = r + + + ....... +
u∈N [v ]      q q            q
4
1442443       4
m − times

p       p        p
< + m = (m + 1) > 1,           >                                   .
q       q        q
Sub case 2: Let v k ∉ N [v ] .
p p          p      p
Then ∑ g (u ) = + + ....... + = (m + 1) > 1.
u∈N [v ]     q q           q      q
4
144 244 3    4
(m +1)−times
Hence, it follows that      ∑ g (u ) ≥ 1,
u∈N [v ]
∀          ∈

Thus is a DF.
This implies that        is not a MDF.
Case 4: Suppose                  .
As in case 1 and 2, we have that
p            1            2
∑[v ] f (u ) = (m + 3) q = (m + 3) m + 1 = 1 + m + 1 > 1,
u∈N
.

p             1
and ∑ f (u ) = (m + 1) = (m + 1)              = 1,
u∈N [v ]               q          m +1
Therefore for all possibilities, we get             ∑ f (u ) ≥ 1, ∀
u∈N [v ]
∈

This implies that is a DF.
Now we check for the minimality of                     .
Define                 by
r, if v = v k ∈ V ,

p
 q , otherwise.

where                .
Since strict inequality holds at a vertex                     of V, it follows that

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME

Then we can show as in case (i) that for                                 and               , we have
p p               p
∑[v ] g (u ) = r + q + q + ....... + q > 1.
u∈N
4
1442443         4
( m + 2 )−times
and for              and              , we have
p p               p
∑[v ] g (u ) = q + q + ....... + q > 1.
u∈ N
4
1442443         4
(m + 3 )− times
Again we can show as in case (ii), that for                                  and             we have
p p               p
∑[v ] g (u ) = r + q + q + ....... + q < 1.
u∈N
4
1442443         4
m − times
and for              and              , we have
p p               p           p            1
∑[v ] g (u ) = q + q + ....... + q = (m + 1) q = (m + 1) m + 1 = 1.
u∈N
4
1442443         4
(m +1)−times
This implies that    ∑ g (u ) < 1,
u∈N [v ]
for some v ∈ V.

So is not a DF.
Since is defined arbitrarily, it follows that there exists no                                such that   is a DF.
Thus is a MDF.

ILLUSTRATION

0                         0

0
0

0       0
0                                            1
0
1                     1

0                 0                                               0       0
1             1

0            0                                   0
0

0            0                     0             0

The function f takes the value 1 for vertices of Cn and value 0 for vertices of Km in each copy.

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REFERENCES

1. Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. (1998), Domination in Graphs: Advanced
Topics, Marcel Dekker, Inc., New York.
2. Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. (1998), Fundamentals of domination in graphs,
Marcel Dekker, Inc., New York .
3. Frucht, R. and Harary, F.(1970), On the corona of Two Graphs, Aequationes Mathematicae,
Volume 4, Issue 3, pp. 322 – 325.
4. Allan, R.B. and Laskar, R.C. (1978), On domination, independent domination numbers of a
graph. Discrete Math., 23, 73 – 76.
5. Cockayne, E.J. and Hedetniemi, S.T. (1977), Towards a theory of domination in graphs.
Networks, 7, 247 – 261.
6. Jeelani Begum, S. (2011), Some studies on dominating functions of Quadratic Residue Cayley
Graphs, Ph. D. thesis, Sri Padmavathi Mahila Visvavidyalayam, Tirupati, Andhra Pradesh,
India.
7. Siva Parvathi, M. (2013), Some studies on dominating functions of Corona Product Graphs, Ph.
D. thesis to be submitted to Sri Padmavathi Mahila Visvavidyalayam, Tirupati, Andhra
Pradesh, India.
8. László Lengyel, “The Role of Graph Transformations in Validating Domain-Specific
Properties”, International Journal of Computer Engineering & Technology (IJCET), Volume 3,
Issue 3, 2012, pp. 406 - 425, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.
9. Syed Abdul Sattar, Mohamed Mubarak.T, Vidya Pv and Appa Rao, “Corona Based Energy
Efficient Clustering in WSN”, International Journal of Advanced Research in Engineering &
Technology (IJARET), Volume 4, Issue 3, 2013, pp. 233 - 242, ISSN Print: 0976-6480, ISSN
Online: 0976-6499.

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