1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME
248
MINIMAL DOMINATING FUNCTIONS OF CORONA PRODUCT GRAPH
OF A CYCLE WITH A COMPLETE GRAPH
1
M.Siva Parvathi, 2
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 ‫ܩ‬ ൌ ‫ܥ‬௡Í ‫ܭ‬௠.
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.
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
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ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 4, Issue 4, July-August (2013), pp. 248-256
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2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME
249
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 ‫ܩ‬ ൌ ‫ܥ‬௡Í ‫ܭ‬௠ and
some results on minimal dominating functions are obtained.
2. CORONA PRODUCT OF ࡯࢔ AND ࡷ࢓
The corona product of a cycle ‫ܥ‬௡ with a complete graph ‫ܭ‬௠ is a graph obtained by taking
one copy of a n – vertex graph ‫ܥ‬௡ and n copies of ‫ܭ‬௠ and then joining the ݅௧௛
vertex of ‫ܥ‬௡ to every
vertex of ݅௧௛
copy of ‫ܭ‬௠ and it is denoted by ‫ܥ‬௡ ‫ܭ‬௠.
We present some properties of the corona product graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ without proofs and the
proofs can be found in Siva Parvathi, M. [7].
Theorem 2.1: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is a connected graph.
Theorem 2.2: The degree of a vertex v in G ൌ ‫ܥ‬௡ ‫ܭ‬௠ is given by
݀ሺ‫ݒ‬ሻ ൌ ൜
݉ ൅ 2 , ݂݅ ‫ݒ‬ ‫א‬ ‫ܥ‬௡,
݉ , ݂݅ ‫ݒ‬ ‫א‬ ‫ܭ‬௠.
Theorem 2.3: The number of vertices and edges in ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is given respectively by
1. |ܸሺ‫ܩ‬ሻ| ൌ ݊ሺ݉ ൅ 1ሻ ,
2. |‫ܧ‬ሺ‫ܩ‬ሻ| ൌ
௡
ଶ
ሺ݉ଶ
൅ ݉ ൅ 2ሻ .
Theorem 2.4: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is non - hamiltonian.
Theorem 2.5: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is eulerian if ݉ is even.
Theorem 2.6: The graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ 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 ‫ܩ‬ ሺܸ, ‫ܧ‬ሻ be a graph. A subset ‫ܦ‬ of ܸ is said to be a dominating set (DS) of ‫ܩ‬ if
every vertex in ܸ – ‫ܦ‬ is adjacent to some vertex in ‫.ܦ‬
A dominating set ‫ܦ‬ is called a minimal dominating set (MDS) if no proper subset of ‫ܦ‬ is a
dominating set of ‫.ܩ‬
Definition: The domination number of ‫ܩ‬ is the minimum cardinality taken over all minimal
dominating sets in ‫ܩ‬ and is denoted by γሺ‫ܩ‬ሻ.
Definition: Let ‫ܩ‬ሺܸ, ‫ܧ‬ሻ be a graph. A function ݂: ܸ ՜ ሾ0,1ሿ is called a dominating function (DF)
of ‫ܩ‬ if ݂ሺܰሾ‫ݒ‬ሿሻ ൌ ( )
[ ]
,1≥∑∈ vNu
uf ݂‫ݎ݋‬ ݄݁ܽܿ ‫ݒ‬∈ܸ.
Here ܰሾ‫ݒ‬ሿ is the closed neighbourhood set of ‫.ݒ‬
Definition: Let ݂ and ݃ be functions from ܸ to [0,1]. We define ݂ ൏ ݃ if ݂ሺ‫ݑ‬ሻ ≤ ݃ሺ‫ݑ‬ሻ for all
‫ݑ‬ ∈ ܸ, with strict inequality for atleast one vertex ‫.ݑ‬

3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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A dominating function ݂ of ‫ܩ‬ is called a minimal dominating function (MDF) if for all
݃ ൏ ݂, ݃ is not a dominating function.
Theorem 3.1: The domination number of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ is n. i.e. ߛሺ‫ܩ‬ሻ ൌ ݊.
Proof: Let ‫ܦ‬ denote a dominating set of ‫.ܩ‬
Case 1: Suppose ‫ܦ‬ contains the vertices of ‫ܥ‬௡ in ‫.ܩ‬
By the definition of ‫,ܩ‬ each vertex in ‫ܥ‬௡ is adjacent to all vertices of ith
copy of ‫ܭ‬௠ . That is the
vertices in ‫ܥ‬௡ dominate the vertices in all copies of ‫ܭ‬௠ respectively. Therefore the vertices of ‫ܦ‬
dominate all vertices of ‫.ܩ‬
Thus ‫ܦ‬ becomes a DS of ‫.ܩ‬ This set is also minimal, because, if we delete one vertex say ‫ݒ‬௞ from ‫,ܦ‬
then the vertices in the kth
copy of ‫ܭ‬௠ are not dominated by any vertex in ‫.ܦ‬
Case 2: Suppose ‫ܦ‬ contains any vertex of each copy of ‫ܭ‬௠ in ‫.ܩ‬
That is |‫|ܦ‬ ൌ ݊. Obviously every vertex in ‫ܭ‬௠ dominates every other vertex in ‫ܭ‬௠ and also a single
vertex of ‫ܥ‬௡ to which it is connected. Therefore the vertices in ‫ܦ‬ dominate all vertices of ‫.ܩ‬ Further
this set is also minimal.
Hence in either case we get ߛሺ‫ܩ‬ሻ ൌ ݊.
Theorem 3.2: Let ‫ܦ‬ be a MDS of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠. Then a function ݂ ‫׷‬ ܸ ՜ ሾ0, 1ሿ defined by
݂ሺ‫ݒ‬ሻ ൌ


 ∈
otherwise.0,
,vif1, D
becomes a MDF of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠.
Proof: Consider the graph ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ with vertex set V.
Let ‫ܦ‬ be a MDS of G. Then by Theorem 3.1, ‫ܦ‬ contains either all the vertices of Cn or any vertex of
each copy of Km in G. For definiteness let ‫ܦ‬ contain the vertices of ‫ܥ‬௡.
Case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬
Then ܰሾ‫ݒ‬ሿ contains m vertices of ‫ܭ‬௠ and three vertices of ‫ܥ‬௡ in ‫.ܩ‬
So
[ ]
( ) .30.......0111 =+++++=
−∈
∑ 43421
timesmvNu
uf
Case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫.ܩ‬
Then ܰሾ‫ݒ‬ሿ contains m vertices of ‫ܭ‬௠ and one vertex of ‫ܥ‬௡ in ‫.ܩ‬
So
[ ]
( ) 10.......01 =+++=
−∈
∑ 43421
timesmvNu
uf .
Therefore for all possibilities, we get
[ ]
( ) ,1≥∑∈
uf
vNu
∀ v ∈ V.
This implies that ݂ is a DF.
Now we check for the minimality of ݂.
Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by
݃ሺ‫ݒ‬ሻ ൌ





∈
∈=
otherwise.0,
},{v-Dvif1,
,vvifr,
k
k D
where 0 ൏ ‫ݎ‬ ൏ 1.
Since strict inequality holds at the vertex ,Dvk ∈ it follows that ݃ ൏ ݂.
Now the following cases arise.
Case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬
Sub case 1: Let ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ.

4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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Then
[ ]
( ) .120.......011 >+=+++++=
−∈
∑ rrug
timesmvNu
43421
Sub case 2: Let ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ.
Then
[ ]
( ) .30.......0111 =+++++=
−∈
∑ 43421
timesmvNu
ug
Case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫.ܩ‬
Sub case 1: Let ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ.
Then
[ ]
( ) .10.......0 <=+++=
−∈
∑ rrug
timesmvNu
43421
Sub case 2: Let ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ.
Then
[ ]
( ) .10.......01 =+++=
−∈
∑ 43421
timesmvNu
ug
This implies that
[ ]
( ) 1<∑∈
ug
vNu
, for some ‫ݒ‬ ∈ ܸ.
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 ݂ሺ‫ݒ‬ሻ ൌ
ଵ
௤ାଵ
, ∀ ‫ݒ‬ ∈ ܸ is a DF of ‫ܩ‬ ൌ
‫ܥ‬௡ ‫ܭ‬௠, if ‫ݍ‬ ൑ ݉. It is a MDF if ‫ݍ‬ ൌ ݉.
Proof: Let V be the vertex set of ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ and ݂ be a function defined as in the hypothesis.
Case 1: Suppose 0 ൏ ‫ݍ‬ ൏ ݉.
Sub case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬
Then ܰሾ‫ݒ‬ሿ contains ݉ vertices of ‫ܭ‬௠ and three vertices of ‫ܥ‬௡ in ‫.ܩ‬
So
[ ]
( )
( )
,1
1
3
1
1
.......
1
1
1
1
3
>
+
+
=
+
++
+
+
+
=
−+
∈
∑ q
m
qqq
uf
timesm
vNu
4444 34444 21
since ‫ݍ‬ ൏ ݉.
Sub case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫.ܩ‬
So
[ ]
( )
( )
,1
1
1
1
1
.......
1
1
1
1
1
>
+
+
=
+
++
+
+
+
=
−+
∈
∑ q
m
qqq
uf
timesm
vNu
4444 34444 21
since ‫ݍ‬ ൏ ݉.
Therefore for all possibilities, we get
[ ]
( ) ,1>∑∈
uf
vNu
∀ v ∈ V.
This implies that ݂ is a DF.
Now we check for the minimality of ݂.
Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by
݃ሺ‫ݒ‬ሻ ൌ





+
∈=
otherwise.,
1q
1
,vvifr, k V
where 0 ൏ ‫ݎ‬ ൏
ଵ
௤ାଵ
.
Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂.
Case (i): Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in G.
Sub case 1: Let ‫ݒ‬௞ ‫א‬ Nሾvሿ.

5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
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Then
[ ]
( )
( )
4444 34444 21
timesm
vNu qqq
rug
−+
∈ +
++
+
+
+
+=∑
2
1
1
.......
1
1
1
1
1
1
3
1
2
1
1
>
+
+
=
+
+
+
+
<
q
m
q
m
q
, since ‫ݍ‬ ൏ ݉.
Sub case 2: Let ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ.
Then
[ ]
( )
( )
,1
1
3
1
1
.......
1
1
1
1
3
>
+
+
=
+
++
+
+
+
=
−+
∈
∑ q
m
qqq
ug
timesm
vNu
4444 34444 21
since q ൏ ݉.
Case (ii): Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in G.
Then we can see similarly as above that for the possibilities ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( ) ,1
1
1
11
1
1
1
.......
1
1
1
1
>
+
+
=+
+
+
<
+
++
+
+
+
+=
−
∈
∑ q
m
q
m
qqqq
rug
timesm
vNu
4444 34444 21
since q ൏ ݉.
and for ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( )
( )
,1
1
1
1
1
.......
1
1
1
1
1
>
+
+
=
+
++
+
+
+
=
−+
∈
∑ q
m
qqq
ug
timesm
vNu
4444 34444 21
since q ൏ ݉.
Hence it follows that
[ ]
( ) ,1≥∑∈
ug
vNu
∀ v ∈ V.
Thus ݃ is a DF.
This implies that ݂ is not a MDF.
Case 2: Suppose ‫ݍ‬ ൌ ݉.
Substituting q = m in Case 1 we can see that for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ , we have
[ ]
( )
( )
.1
1
2
1
1
3
1
3
1
1
.......
1
1
1
1
3
>
+
+=
+
+
=
+
+
=
+
++
+
+
+
=
−+
∈
∑ mm
m
q
m
qqq
uf
timesm
vNu
4444 34444 21
and for ‫ݒ‬ ‫א‬ ‫ܭ‬௠, we have
[ ]
( )
( )
.1
1
1
1
1
1
1
.......
1
1
1
1
1
=
+
+
=
+
+
=
+
++
+
+
+
=
−+
∈
∑ m
m
q
m
qqq
uf
timesm
vNu
4444 34444 21
Therefore for all possibilities, we get
[ ]
( ) ,1≥∑∈
uf
vNu
∀ v ∈ V.
This implies that ݂ is a DF.
Now we check for the minimality of ݂.
Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by
݃ሺ‫ݒ‬ሻ ൌ




+
∈=
otherwise.,
1m
1
,vvifr, k V
where 0 ൏ ‫ݎ‬ ൏
ଵ
௠ାଵ
.
Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂.
Then we can see as in case (i), that for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ , we have
[ ]
( )
( )
44444 344444 21
timesm
vNu mmm
rug
−+
∈ +
++
+
+
+
+=∑
2
1
1
.......
1
1
1
1
1
1
3
1
2
1
1
>
+
+
=
+
+
+
+
<
m
m
m
m
m

6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
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253
and for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( )
( )
.1
1
3
1
1
.......
1
1
1
1
3
>
+
+
=
+
++
+
+
+
=
−+
∈
∑ m
m
mmm
ug
timesm
vNu 44444 344444 21
Similarly we can show as in case (ii) that for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( ) .1
11
1
1
1
.......
1
1
1
1
=
+
+
+
<
+
++
+
+
+
+=
−
∈
∑ m
m
mmmm
rug
timesm
vNu 44444 344444 21
and for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( )
( )
.1
1
1
1
1
.......
1
1
1
1
1
=
+
+
=
+
++
+
+
+
=
−+
∈
∑ m
m
mmm
ug
timesm
vNu 44444 344444 21
This implies that
[ ]
( ) 1<∑∈
ug
vNu
, 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 ݂ሺ‫ݒ‬ሻ ൌ
௣
௤
, ∀ ‫ݒ‬ ∈ ܸ where p = min (m,
n) and q = max ( m, n ) is a DF if
௣
௤
൒
ଵ
௠ାଵ
. Otherwise it is not a DF. Also it becomes MDF
if
௣
௤
ൌ
ଵ
௠ାଵ
.
Proof: Let ‫ܩ‬ ൌ ‫ܥ‬௡ ‫ܭ‬௠ be the given graph with vertex set ܸ.
Let ݂ ‫׷‬ ܸ → ሾ 0, 1 ሿ be defined by ݂ሺ‫ݒ‬ሻ ൌ
௣
௤
, ∀ ‫ݒ‬ ∈ܸ , where p = min(m, n) and q=max ( m, n).
Clearly
௣
௤
൐ 0.
Case 1: Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫ܩ‬.
Then
[ ]
( )
( )
( ) .3.......
3
q
p
m
q
p
q
p
q
p
uf
timesm
vNu
+=+++=
−+
∈
∑
444 3444 21
Case 2: Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in G.
Then
[ ]
( )
( )
( ) .1.......
1
q
p
m
q
p
q
p
q
p
uf
timesm
vNu
+=+++=
−+
∈
∑
444 3444 21
From the above two cases, we observe that ݂ is a DF if
௣
௤
൒
ଵ
௠ାଵ
.
Otherwise ݂ is not a DF.
Case 3: Suppose
௣
௤
>
ଵ
௠ାଵ
.
Clearly ݂ is a DF.
Now we check for the minimality of ݂.
Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by
݃ሺ‫ݒ‬ሻ ൌ




 ∈=
otherwise.,
q
p
,vvifr, k V
where 0 ൏ ‫ݎ‬ ൏
௣
௤
.

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Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂.
Case (i): Let ‫ݒ‬ ‫א‬ ‫ܥ‬௡ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 2 in ‫ܩ‬.
Sub case 1: Let ‫ݒ‬௞ ‫א‬ Nሾvሿ.
Then
[ ]
( )
( )
444 3444 21
timesm
vNu q
p
q
p
q
p
rug
−+
∈
++++=∑
2
.......
( ) ( ) ,132 >+=++<
q
p
m
q
p
m
q
p
since
௣
௤
>
ଵ
௠ାଵ
.
Sub case 2: Let [ ]vNvk ∉ .
Then
[ ]
( )
( )
( ) .13.......
3
>+=+++=
−+
∈
∑ q
p
m
q
p
q
p
q
p
ug
timesm
vNu
444 3444 21
Case (ii): Let ‫ݒ‬ ‫א‬ ‫ܭ‬௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ in ‫ܩ‬.
Sub case 1: Let ‫ݒ‬௞ ‫א‬ Nሾvሿ.
Then
[ ]
( )
444 3444 21
timesm
vNu q
p
q
p
q
p
rug
−
∈
++++=∑ .......
( ) ,11 >+=+<
q
p
m
q
p
m
q
p
since
௣
௤
>
ଵ
௠ାଵ
.
Sub case 2: Let [ ]vNvk ∉ .
Then
[ ]
( )
( )
( ) .11.......
1
>+=+++=
−+
∈
∑ q
p
m
q
p
q
p
q
p
ug
timesm
vNu
444 3444 21
Hence, it follows that
[ ]
( ) ,1≥∑∈
ug
vNu
∀ v ∈ V.
Thus ݃ is a DF.
This implies that ݂ is not a MDF.
Case 4: Suppose
௣
௤
ൌ
ଵ
௠ାଵ
.
As in case 1 and 2, we have that
[ ]
( ) ( ) ( ) ,1
1
2
1
1
1
33 >
+
+=
+
+=+=∑∈ mm
m
q
p
muf
vNu
݂݅ ‫ݒ‬ ‫א‬ ‫ܥ‬௡ .
and
[ ]
( ) ( ) ( ) ,1
1
1
11 =
+
+=+=∑∈ m
m
q
p
muf
vNu
݂݅ ‫ݒ‬ ‫א‬ ‫ܭ‬௠.
Therefore for all possibilities, we get
[ ]
( ) ,1≥∑∈
uf
vNu
∀ v ∈ V.
This implies that ݂ is a DF.
Now we check for the minimality of ݂.
Define ݃ ‫׷‬ ܸ ՜ ሾ0, 1ሿ by
݃ሺ‫ݒ‬ሻ ൌ




 ∈=
otherwise.,
q
p
,vvifr, k V
where 0 ൏ ‫ݎ‬ ൏
௣
௤
.
Since strict inequality holds at a vertex ‫ݒ‬௞ of V, it follows that ݃ ൏ ݂.

8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME
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Then we can show as in case (i) that for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( )
( )
.1.......
2
>++++=
−+
∈
∑
444 3444 21
timesm
vNu q
p
q
p
q
p
rug
and for ‫ݒ‬ ‫א‬ ‫ܥ‬௡ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( )
( )
.1.......
3
>+++=
−+
∈
∑
444 3444 21
timesm
vNu q
p
q
p
q
p
ug
Again we can show as in case (ii), that for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫א‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( ) .1....... <++++=
−
∈
∑
444 3444 21
timesm
vNu q
p
q
p
q
p
rug
and for ‫ݒ‬ ‫א‬ ‫ܭ‬௠ and ‫ݒ‬௞ ‫ב‬ ܰሾ‫ݒ‬ሿ, we have
[ ]
( )
( )
( ) ( ) .1
1
1
11.......
1
=
+
+=+=+++=
−+
∈
∑ m
m
q
p
m
q
p
q
p
q
p
ug
timesm
vNu
444 3444 21
This implies that
[ ]
( ) 1<∑∈
ug
vNu
, 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
1
1
11
1
0
0
0
0
0
0
0
0
00
00
0 0
0 0
0
0
00
࡯૞ ٖ ࡷ૝
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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