1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
TECHNOLOGY (IJCET)
ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 5, Issue 1, January (2014), pp. 21-27
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IJCET
©IAEME
ROMAN DOMINATING AND TOTAL DOMINATING FUNCTIONS OF
CORONA PRODUCT GRAPH OF A PATH WITH A STAR
1
Makke Siva Parvathi,
2
Bommireddy Maheswari
1
2
Dept. of Mathematics, K.R.K. Govt. Degree College, Addanki-523201, Andhra Pradesh, India
Dept. of Applied Mathematics, S.P.Women’s University, Tirupat-517502, Andhra Pradesh, India
ABSTRACT
In recent years ‘Domination in Graphs’ received much attention and it is an important branch
of graph theory. Haynes et al. [1, 2] have given an extensive overview on domination in graphs and
related topics in the two books. Recently dominating functions in domination theory have received
much attention. In this paper we present some results on minimal Roman and total Roman
dominating functions of corona product graph of a path with a star.
Key Words: Corona Product, Roman dominating function, Total Roman dominating function.
Subject Classification: 68R10
1. INTRODUCTION
Graph theory is one of the most flourishing branches of modern mathematics and finds
application in various branches of Science & Technology. Domination in graphs has a wide range of
applications to many fields like Engineering, Communication Networks, Social sciences, linguistics,
physical sciences and many others. Allan, R.B. and Laskar, R.[3], Cockayne, E.J. and Hedetniemi,
S.T. [4] have studied various domination parameters of graphs.
The Roman dominating function of a graph G was defined by Cockayne et al. [5]. The
definition of a Roman dominating function was motivated by an article in Scientific American by Ian
Stewart [6] entitled “Defend the Roman Empire!” and suggested by even earlier by ReVelle[7].
Frucht and Harary [8] 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.
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2. International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
The authors have studied some dominating functions of corona product graph of a cycle with
a complete graph [9] and published papers on minimal dominating functions, some variations of Y –
dominating functions, Y – total dominating functions , total dominating functions and convexity of
dominating and total dominating functions[10,11,12,13,14].
In this paper we discuss some results on Roman dominating functions and total Roman
dominating functions of corona product graph of a path with a star.
2. CORONA PRODUCT OF ࡼ࢔ AND ࡷ૚,࢓
The corona product of a path ܲ௡ with star ‫ܭ‬ଵ,௠ is a graph obtained by taking one copy of a
݊ – vertex path ܲ and n copies of ‫ܭ‬ଵ,௠ and then joining the ݅௧௛ vertex of ܲ to every vertex of ݅ ௧௛
௡
௡
copy of ‫ܭ‬ଵ,௠ and it is denoted by ܲ௡ ‫ܭ‬ଵ,௠ .
We require the following theorem whose proof can be found in Siva Parvathi, M. [8].
Theorem 2.1: The degree of a vertex ‫ݒ‬௜ in ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ is given by
݉ ൅ 3,
݂݅ ‫ݒ‬௜ ‫ܲ א‬௡ ܽ݊݀ 2 ൑ ݅ ൑ ሺ݊ െ 1ሻ,
‫ۓ‬
݉ ൅ 2,
݂݅ ‫ݒ‬௜ ‫ܲ א‬௡ ܽ݊݀
݅ ൌ 1 ‫,݊ ݎ݋‬
݀ሺ‫ݒ‬௜ ሻ ൌ
݂݅ ‫ݒ‬௜ ‫ܭ א‬ଵ,௠ ܽ݊݀ ‫ݒ‬௜ ݅‫,݊݋݅ݐ݅ݐݎܽ݌ ݐݏݎ݂݅ ݊݅ ݏ‬
‫ ݉ ۔‬൅ 1,
2,
݂݅ ‫ݒ‬௜ ‫ܭ א‬ଵ,௠ ܽ݊݀ ‫ݒ‬௜ ݅‫.݊݋݅ݐ݅ݐݎܽ݌ ݀݊݋ܿ݁ݏ ݊݅ ݏ‬
‫ە‬
3. ROMAN DOMINATING FUNCTIONS
In this section we discuss Roman dominating functions and minimal Roman dominating
functions of the graph ‫ ܩ‬ൌ ܲ Í ‫ܭ‬ଵ,௠ . First we recall the definitions of Roman dominating function
௡
of a graph.
Definition: Let ‫ ܩ‬ሺ ܸ, ‫ ܧ‬ሻ be a graph. A function ݂ ‫ ܸ ׷‬՜ ሼ 0, 1, 2 ሽ is called a Roman
dominating function (RDF) of ‫ ܩ‬if ݂ሺܰሾ‫ݒ‬ሿሻ ൌ
∑ f (u ) ≥ 1, ݂‫݄ܿܽ݁ ݎ݋‬
‫ ܸ ∈ ݒ‬and satisfying the
condition that every vertex ‫ ݑ‬for which ݂ሺ‫ݑ‬ሻ ൌ 0 is adjacent to at least one vertex ‫ ݒ‬for which
݂ ሺ‫ݒ‬ሻ ൌ 2.
A Roman dominating function ݂ of ‫ ܩ‬is called a minimal Roman dominating function
(MRDF) if for all ݃ ൏ ݂, ݃ is not a Roman dominating function.
Theorem 3.1: A function ݂ ‫ ܸ ׷‬՜ ሼ 0, 1, 2 ሽ defined by
2, for v whose degree is m + 1 in each copy of K 1, m in G,
݂ ሺ‫ ݒ‬ሻ ൌ 
0, otherwise.
is a minimal Roman dominating function of ‫ ܩ‬ൌ ܲ௡ Í‫ܭ‬ଵ,௠ .
Proof: Let ݂ be a function defined as in the hypothesis.
Case 1: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 3 in ‫.ܩ‬
௡
Then ܰሾ‫ݒ‬ሿ contains ݉ ൅ 1 vertices of ‫ܭ‬ଵ,௠ and three vertices of ܲ in ‫.ܩ‬
௡


So ∑ f (u ) = 0 + 0 + 0 + 2 + 0 + ....... + 0  = 2.
14 4
2 3
u∈N [v ]
m −times


Case 2: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬
௡
Then ܰሾ‫ݒ‬ሿ contains ݉ ൅ 1 vertices of ‫ܭ‬ଵ,௠ and two vertices of ܲ in ‫.ܩ‬
௡


So ∑ f (u ) = 0 + 0 + 2 + 0 + ....... + 0 = 2.
14 4
2 3
u ∈ N [v ]
m −times


Case 3: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in ‫.ܩ‬
u∈N [v ]
22

3. International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
Then ܰሾ‫ݒ‬ሿ contains ݉ ൅ 1 vertices of ‫ܭ‬ଵ,௠ and one vertex of ܲ in ‫.ܩ‬
௡


So ∑ f (u ) = 0 + 2 + 0 + ....... + 0 = 2.
14 4
2 3
u ∈ N [v ]
m −times


Case 4: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ 2 in ‫.ܩ‬
Then ܰሾ‫ݒ‬ሿ contains two vertices of ‫ܭ‬ଵ,௠ and one vertex of ܲ௡ in ‫.ܩ‬
So ∑ f (u ) = 0 + 2 + 0 = 2.
u∈N [v ]
Therefore for all possibilities, we get
∑ f (u ) > 1, ∀
v ∈ V.
Let ‫ ݑ‬be any vertex in ‫ ܩ‬such that ݂ሺ‫ݑ‬ሻ ൌ 0. Then ‫ ܲ ∈ ݑ‬or ‫ܭ ∈ ݑ‬ଵ,௠ such that ݀ሺ‫ݑ‬ሻ ൌ 2. Let
௡
‫ ݑ ് ݒ‬be a vertex in ‫ ܩ‬such that ݂ሺ‫ݒ‬ሻ ൌ 2. Then ‫ܭ ∈ݒ‬ଵ,௠ such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1.
We now show that ‫ ݑ‬is adjacent to ‫ . ݒ‬If ‫ܲ ∈ ݑ‬௡ , then ‫ ݑ‬is adjacent to ‫ , ݒ‬since every vertex in ܲ௡
is adjacent to every vertex in the corresponding copy of ‫ܭ‬ଵ,௠ .
Also if ‫ܭ ∈ ݑ‬ଵ,௠ then ‫ ݑ‬is adjacent to ‫ܭ ∈ ݒ‬ଵ,௠ , since ‫ ݑ‬and ‫ ݒ‬are in different partitions of ‫ܭ‬ଵ,௠ .
This implies that ݂ is a RDF.
Now we check for the minimality of ݂.
Define g : V → {0, 1, 2} by
u∈N [v ]
1,

݃ሺ‫ ݒ‬ሻ ൌ 2,
0,

for v whose degree is m + 1 in the i th copy of K 1,m in G,
for v whose degree is m + 1 in (n - 1) copies of K 1, m in G,
otherwise.
Since strict inequality follows at the vertex v ∈ V , it follows that ݃ ൏ ݂.
Case (i): Let ‫ ܲ א ݒ‬be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3 in ‫.ܩ‬
௡
Sub case 1: Let ‫ ݒ‬be in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫.ܩ‬


Then ∑ g (u ) = 0 + 0 + 0 + 1 + 0 + ....... + 0 = 1.
14 4
2 3
u∈N [v ]
m −times


Sub case 2: Let ‫ ݒ‬be not in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫.ܩ‬


Then ∑ g (u ) = 0 + 0 + 0 + 2 + 0 + ....... + 0  = 2.
14 4
2 3
u ∈ N [v ]
m − times


Case (ii): Let ‫ܲ א ݒ‬௡ be such that ݀ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬
Sub case 1: Let ‫ ݒ‬be in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫.ܩ‬


Then ∑ g (u ) = 0 + 0 + 1 + 0 + ....... + 0 = 1.
14 4
2 3
u ∈ N [v ]
m − times


Sub case 2: Let ‫ ݒ‬be not in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫.ܩ‬


Then ∑ g (u ) = 0 + 0 + 2 + 0 + ....... + 0  = 2.
14 4
2 3
u∈N [v ]
m −times


ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in ‫.ܩ‬
Case (iii): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀
Sub case 1: Let ‫ ݒ‬be in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫.ܩ‬


Then ∑ g (u ) = 0 + 1 + 0 + ....... + 0 = 1.
14 4
2 3
u ∈ N [v ]
m − times


௧௛
Sub case 2: Let ‫ ݒ‬be not in the ݅ copy of ‫ܭ‬ଵ,௠ in ‫.ܩ‬
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4. International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
Then


∑[ ] g (u ) = 0 + 2 + 0 +4243  = 2.
1 ....... + 0

Case (iv): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ 2 in ‫.ܩ‬
Sub case 1: Let ‫ ݒ‬be in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫.ܩ‬
Then ∑ g (u ) = 0 + 1 + 0 = 1.
u∈ N v

m − times
u∈N [v ]
Sub case 2: Let ‫ ݒ‬be not in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫. ܩ‬
Then ∑ g (u ) = 0 + 2 + 0 = 2.
u∈N [v ]
This implies that
∑ g (u ) ≥ 1,
∀ v ∈ V.
u∈N [v ]
i.e. ݃ is a DF. But ݃ is not a RDF, since the RDF definition fails in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ in ‫. ܩ‬
Because the vertex ‫ ݑ‬in the ݅ ௧௛ copy of ‫ܭ‬ଵ,௠ of ‫ ܩ‬for which ݂ሺ‫ݑ‬ሻ ൌ 0 is adjacent to a vertex
‫ ݒ‬for which ݂ ሺ‫ݒ‬ሻ ൌ 1.
Therefore ݂ is a MRDF. ‫ז‬
4. TOTAL ROMAN DOMINATING FUNCTIONS
In this section we introduce the concept of total Roman dominating function of a graph G.
Some results on minimal total Roman dominating function of ‫ ܩ‬ൌ ܲ௡ ‫ܭ‬ଵ,௠ are obtained. First we
define total Roman dominating function of a graph.
Definition: Let ‫ ܩ‬ሺ ܸ, ‫ ܧ‬ሻ be a graph. A function ݂ ‫ ܸ ׷‬՜ ሼ 0, 1, 2ሽ is called a total Roman
dominating function (TRDF) of ‫ ܩ‬if ݂ ሺܰሺ‫ݒ‬ሻሻ ൌ ∑ f (u ) ≥ 1, ݂‫ ܸ ∈ ݒ ݄ܿܽ݁ ݎ݋‬and satisfying
u∈N ( v )
the condition that every vertex ‫ ݑ‬for which ݂ሺ‫ݑ‬ሻ ൌ 0 is adjacent to at least one vertex ‫ ݒ‬for which
݂ ሺ‫ݒ‬ሻ ൌ 2.
A total Roman dominating function ݂ of ‫ ܩ‬is called a minimal total Roman dominating
function (MTRDF) if for all ݃ ൏ ݂, ݃ is not a total Roman dominating function.
Theorem 4.1: A function f : V → { 0, 1, 2} defined by
2, for the vertices of Pn in G,
݂ሺ‫ݒ‬ሻ ൌ 
0, otherwise .
is a minimal Total Roman Dominating Function of ‫ ܩ‬ൌ ܲ ‫ܭ‬ଵ,௠ .
௡
Proof: Let ݂ be a function defined as in the hypothesis.
Case 1: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 3 in ‫.ܩ‬
௡
Then ܰሺ‫ݒ‬ሻ contains ݉ ൅ 1 vertices of ‫ܭ‬ଵ,௠ and two vertices of ܲ௡ in ‫.ܩ‬


So ∑ f (u ) = 2 + 2 + 0 + 04........ + 0 = 4.
14 +244
3
u∈N ( v )


( m +1) −times


Case 2: Let ‫ ܲ א ݒ‬be such that ݀ ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in ‫.ܩ‬
௡
Then ܰሺ‫ݒ‬ሻ contains ݉ ൅ 1 vertices of ‫ܭ‬ଵ,௠ and one vertex of ܲ௡ in ‫.ܩ‬


So ∑ f (u ) = 2 + 0 + 04........ + 0 = 2.
+244
14
3
u∈N ( v )


( m +1) −times


ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in ‫.ܩ‬
Case 3: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀
Then ܰሺ‫ݒ‬ሻ contains ݉ vertices of ‫ܭ‬ଵ,௠ whose degree is 2 and one vertex of ܲ in ‫.ܩ‬
௡
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5. International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
So


∑ f (u ) = 2 + 0 +44244 0 = 2.
1 0 + ........ +
3
m −times


Case 4: Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ 2 in ‫.ܩ‬
Then ܰሺ‫ݒ‬ሻ contains one vertex of ‫ܭ‬ଵ,௠ whose degree is ݉ ൅ 1 and one vertex of ܲ௡ in ‫.ܩ‬
So ∑ f (u ) = 2 + 0 = 2.
u∈N ( v )
u∈N ( v )
∑ f (u ) > 1,
Therefore
∀ v ∈ V.
u∈N ( v )
And every vertex ‫ ݑ‬for which ݂ ሺ‫ ݑ‬ሻ ൌ 0 is in ‫ܭ‬ଵ,௠ and every such vertex is adjacent to a
corresponding vertex say ‫ ݒ‬in ܲ௡ for which ݂ሺ‫ݒ‬ሻ ൌ 2.
This implies that ݂ is a TRDF.
Now we check for the minimality of ݂ .
Define g : V → { 0, 1, 2 } by
1, for v = v k ∈ Pn in G,

݃ሺ‫ ݒ‬ሻ ൌ 2, for v ∈ Pn - {v k } in G,
0, otherwise.

Case (i): Let ‫ ܲ א ݒ‬be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 3 in G.
௡
vk ∈ N (v) .
Sub case 1: Let

Then

∑ g (u ) = 1 + 2 + 0 +44244 0 = 3.
1 0 + ........ +
3


u∈N ( v )


( m+1)−times
Sub case 2: Let vk ∉ N (v) .

Then

∑ g (u ) = 2 + 2 + 0 +44244 0 = 4.
1 0 + ........ +
3


(m +1)−times


Case (ii): Let ‫ܲ א ݒ‬௡ be such that ݀ሺ‫ ݒ‬ሻ ൌ ݉ ൅ 2 in G.
Sub case 1: Let vk ∈ N (v) .
u∈N ( v )

Then

∑ g (u ) = 1 + 0 +44244 0 = 1.
1 0 + ........ +
3

(m +1)−times

Sub case 2: Let vk ∉ N (v) .
u∈N ( v )

Then



∑ g (u ) = 2 + 0 +44244 0 = 2.
1 0 + ........ +
3


Case (iii): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ሺ‫ݒ‬ሻ ൌ ݉ ൅ 1 in G.
Sub case 1: Let vk ∈ N (v) .
u∈N ( v )
Then


(m +1)−times


∑ g (u ) = 1 + 0 +44244 0 = 1.
1 0 + ........ +
3
u∈N ( v )

m −times

Sub case 2: Let vk ∉ N (v) .
Then


∑ g (u ) = 2 + 0 +44244 0 = 2.
1 0 + ........ +
3
m − times


Case (iv): Let ‫ܭ א ݒ‬ଵ,௠ be such that ݀ ሺ‫ ݒ‬ሻ ൌ 2 in G.
u∈N ( v )
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6. International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
Sub case 1: Let vk ∈ N (v) .
Then
∑ g (u ) = 1 + 0 = 1.
u∈N ( v )
Sub case 2: Let vk ∉ N (v) .
Then
∑ g (u ) = 2 + 0 = 2.
∀ v ∈ V.
u∈N ( v )
Therefore for all possibilities, we get
∑ g (u ) ≥ 1,
i.e. ݃ is a TDF. But ݃ is not a TRDF, since the TRDF definition fails in the ݇ ௧௛ copy of ‫ܭ‬ଵ,௠
in G because the vertex ‫ ݑ‬in the ݇ ௧௛ copy of ‫ܭ‬ଵ,௠ of G for which ݂ሺ‫ݑ‬ሻ ൌ 0 is adjacent to a
vertex ‫ݒ‬௞ for which ݂ ሺ‫ݒ‬௞ ሻ ൌ 1.
Therefore ݂ is a MTRDF. ‫ז‬
u∈N ( v )
5. ILLUSTRATION
0
0
2
0
0
0
2
2
0
0
0
0
2
2
0
0
0
0
0
0
0
0
2
0
0
2
2
0
0
0
0
ࡳ ൌ ࡼૡ ٖ ࡷ૚,૛
0
The function ‘f’ takes the value 2 for the vertices of ࡼૡ and the value 0 for the remaining
vertices in G.
6. 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] Allan, R.B. and Laskar, R.C. (1978) – On domination, independent domination numbers of a
graph Discrete Math., 23, pp.73 – 76.
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7. International Journal of Computer Engineering and Technology (IJCET), ISSN 09766367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 1, January (2014), © IAEME
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Cockayne, E.J. and Hedetniemi, S.T. (1977) - Towards a theory of domination in graphs.
Networks, 7, pp.247 – 261.
Cockayne, E.J., Dreyer, P.A., Hedetniemi, S.M. and Hedetniemi, S.T. (2004) - Roman
domination in graphs, Discrete Math., 278, 11 – 22.
Ian Stewart (1999) - Defend the Roman Empire!, Scientific American, 281 (6), 136 – 139.
ReVelle, C.S. and Rosing, K.E. (2000) - Defendens imperium romanum: a classical problem
in military strategy, Amer.Math.Monthly, 107(7), 585 – 594.
Frucht, R. and Harary, F. (1970) - On the corona of Two Graphs, Aequationes
Mathematicae, Volume 4, Issue 3, pp.322 – 325.
Siva Parvathi, M – Some studies on dominating functions of corona product graphs, Ph.D
thesis, Sri Padmavati Mahila Visvavidyalayam, Tirupati, Andhra Pradesh, India. 2013.
Siva Parvathi, M and Maheswari, B . - Minimal Dominating Functions of Corona Product
Graph of a Cycle with a Complete Graph - International Journal of Computer Engineering &
Technology, Volume 4, Issue 4, 2013, pp. 248 – 256.
Siva Parvathi, M and Maheswari, B. - Some variations of Y-Dominating Functions of Corona
Product Graph of a Cycle with a Complete Graph - International Journal of Computer
Applications, Volume 81, Issue 1, 2013, pp. 16 – 21.
Siva Parvathi, M and Maheswari, B. - Some variations of Total Y-Dominating Functions of
Corona Product Graph of a Cycle with a Complete Graph - Fire Journal Science and
Technology (accepted).
Siva Parvathi, M and Maheswari, B. - Minimal total dominating functions of corona product
graph of a cycle with a complete graph – International Journal of Applied Information
Systems (communicated).
Siva Parvathi, M and Maheswari, B. - Convexity of minimal dominating and total dominating
functions of corona product graph of a cycle with a complete graph – International Journal of
Computer Applications(communicated).
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.
M.Manjuri and B.Maheswari - Strong Dominating Sets of Strong Product Graph of Cayley
Graphs with Arithmetic Graphs - International Journal of Computer Engineering &
Technology, Volume 4, Issue 6 (2013), pp. 136 - 144.
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