1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
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ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
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ISSN 0976 – 6375(Online)
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STRONG DOMINATING SETS OF STRONG PRODUCT GRAPH OF
CAYLEY GRAPHS WITH ARITHMETIC GRAPHS
M.Manjuri
and B.Maheswari
Department of Applied Mathematics, S.P.Women’s University, Tirupati,
Andhra Pradesh, India.
ABSTRACT
Graph Theory has been realized as one of the most flourishing branches of modern
Mathematics finding widest applications in all most all branches of Sciences, Social Sciences,
Engineering, Computer Science, etc. Number Theory is one of the oldest branches of Mathematics,
which inherited rich contributions from almost all greatest mathematicians, ancient and modern.
Product of graphs are introduced in Graph Theory very recently and developing rapidly. In
this paper, we consider strong product graphs of Cayley graphs with Arithmetic graphs and present
strong domination parameter of these graphs.
Key words: Euler Totient Cayley graph, Arithmetic ܸ graph, strong Product graph,
௡
Strong domination.
Subject Classification: 68R10
1. INTRODUCTION
Domination in graphs is a flourishing area of research at present. Dominating sets play an
important role in practical applications, such as logistics and networks design, mobile computing,
resource allocation and telecommunication etc. Cayley graphs are excellent models for
interconnection networks, investigated in connection with parallel processing and distributed
computation.
Nathanson [1] was the pioneer in introducing the concepts of Number Theory, particularly,
the ‘Theory of Congruences’ in Graph Theory, thus paved the way for the emergence of a new
class of graphs, namely “Arithmetic Graphs”. Cayley Graphs are another class of graphs associated
with elements of a group. If this group is associated with some Arithmetic function then the Cayley
136

2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
graph becomes an Arithmetic graph. The Cayley graph associated with Euler totient function is
called an Euler totient Cayley graph .
The strong product graphs are studied by Nesetril [2]. Occasionally it is also called as strong
direct product or symmetric composition.
STRONG PRODUCT GRAPH
If ‫ܩ‬ଵ and ‫ܩ‬ଶ are two simple graphs with their vertex sets as ܸଵ ൌ ሼ‫ݑ‬ଵ , ‫ݑ‬ଶ , … , ‫ݑ‬௟ ሽ and
ܸଶ ൌ ሼ‫ݒ‬ଵ , ‫ݒ‬ଶ , … , ‫ݒ‬௠ ሽ respectively then the strong product of these two graphs denoted by the
‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is defined as the graph whose vertex set is ܸଵ ൈ ܸଶ, and any two distinct vertices
ሺ‫ݑ‬ଵ , ‫ݒ‬ଵ ሻ and ሺ‫ݑ‬ଶ , ‫ݒ‬ଶ ሻ of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ are adjacent if
(i) ‫ݑ‬ଵ ൌ ‫ݑ‬ଶ and ‫ݒ‬ଵ ‫ݒ‬ଶ ‫ܧ א‬ሺ‫ܩ‬ଶ ሻ
or
(ii) ‫ݑ‬ଵ ‫ݑ‬ଶ ‫ܧ א‬ሺ‫ܩ‬ଵ ሻ and
or
‫ݒ‬ଵ ൌ ‫ݒ‬ଶ
(iii) ‫ݑ‬ଵ ‫ݑ‬ଶ ‫ܧ א‬ሺ‫ܩ‬ଵ ሻ and ‫ݒ‬ଵ ‫ݒ‬ଶ ‫ܧ א‬ሺ‫ܩ‬ଶ ሻ.
2. EULER TOTIENT CAYLEY GRAPH
For any positive integer ݊, let ܼ௡ ൌ ሼ0,1,2, … . . ݊ െ 1ሽ. Then ሺܼ௡ , ۩ሻ, where, ۩ is addition
modulo ݊, is an abelian group of order ݊. The number of positive integers less than ݊ and
relatively prime to ݊ is denoted by ߮ሺ݊ሻ and is called Euler totient function. Let ܵ denote the set of
all positive integers less than ݊ and relatively prime to ݊ .
That is ܵ ൌ ሼ‫ 1 /ݎ‬൑ ‫ ݎ‬൏ ݊ and GCD ሺ ‫ ݊ ,ݎ‬ሻ ൌ 1 ሽ. Then |ܵ| ൌ ߮ሺ݊ሻ.
Now we define Euler totient Cayley graph as follows.
The Euler totient Cayley graph ‫ ܩ‬ሺܼ௡ , ߮ሻ is defined as the graph whose vertex set ܸ is given
by ܼ௡ ൌ ሼ0,1,2, … . ݊ െ 1ሽ and the edge set is ‫ ܧ‬ൌ ሼሺ‫ݕ ,ݔ‬ሻ⁄‫ ݔ‬െ ‫ ܵ א ݕ‬or ‫ ݕ‬െ ‫ܵ א ݔ‬ሽ.
Some properties of Euler totient Cayley graphs and enumeration of Hamilton cycles and
triangles can be found in Madhavi [3].
The Euler totient Cayley graph ‫ ܩ‬ሺܼ௡ , ߮) is a complete graph if ݊ is a prime and it is ߔሺ݊ሻ െ
regular.
The strong domination parameter of these graphs are studied by the authors [4] and we
require the following results and we present them without proofs.
Theorem 2.1: If ݊ is a prime, then the strong domination number of ‫ܩ‬ሺܼ௡ , ߮ሻ is 1.
Theorem 2.2: If ݊ ൌ 2‫ ݌‬where ‫ ݌‬is an odd prime, then the strong domination number of ‫ܩ‬ሺܼ௡ , ߮ሻ
is 2.
ఈ
ఈ
ఈ
Theorem 2.3: Suppose ݊ is neither a prime nor 2‫ .݌‬Let ݊ ൌ ‫݌‬ଵ భ ‫݌‬ଶ మ … … . . ‫݌‬௞ ೖ , where
‫݌‬ଵ , ‫݌‬ଶ , … . . , ‫݌‬௞ are primes and ߙଵ , ߙଶ , … . , ߙ௞ are integers ൒ 1. Then the strong domination number
of ‫ ܩ‬ሺܼ௡ , ߮ሻ is ߣ ൅ 1, where ߣ is the length of the longest stretch of consecutive integers in ܸ, each
of which shares a prime factor with ݊.
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3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
3. ARITHMETIC ࢂ࢔ GRAPH
ఈ
ఈ
ఈ
Let ݊ be a positive integer such that ݊ ൌ ‫݌‬ଵ భ ‫݌‬ଶ మ … … . . ‫݌‬௞ ೖ . Then the Arithmetic ܸ graph
௡
denoted by ‫ܩ‬ሺܸ ሻ is defined as the graph whose vertex set consists of the divisors of ݊ and two
௡
‫ ܦܥܩ‬ሺ‫ݒ ,ݑ‬ሻ ൌ ‫݌‬௜ , for some prime divisor
vertices ‫ ݒ ,ݑ‬are adjacent in ‫ܩ‬ሺܸ ሻ graph if and only if
௡
‫݌‬௜ of ݊.
In this graph vertex 1 becomes an isolated vertex. Hence we consider ‫ܩ‬ሺܸ ሻ graph without
௡
vertex 1 as the contribution of this isolated vertex is nothing when we study the domination
parameters.
Clearly, ‫ܩ‬ሺܸ ሻ graph is a connected graph. Because if ݊ is a prime, then ‫ܩ‬ሺܸ ሻ graph
௡
௡
consists of a single vertex. Hence it is a connected graph. In other cases, by the definition of
adjacency in ‫ܩ‬ሺܸ ሻ , there exist edges between prime numbers, their prime powers and also to their
௡
prime products. Therefore each vertex of ‫ܩ‬ሺܸ ሻ is connected to some vertex in ‫ܩ‬ሺܸ ሻ .
௡
௡
The strong domination parameter of these graphs are studied by the authors [4] and we
require the following result and we present it without proof.
ఈ
ఈ
ఈ
Theorem 3.1: If ݊ ൌ ‫݌‬ଵ భ ‫݌‬ଶ మ … … . . ‫݌‬௞ ೖ , where ‫݌‬ଵ , ‫݌‬ଶ , … ‫݌‬௞ are primes and ߙଵ , ߙଶ , … . ߙ௞ are
integers ൒ 1, then the strong domination number of ‫ ܩ‬ሺܸ ሻ is given by
௡
݇െ1
if α୧ ൌ 1 for more than one i,
ߛ௦ ൫‫ܩ‬ሺܸ ሻ൯ ൌ ቄ
௡
݇
otherwise.
where ݇ is the core of ݊.
4. STRONG PRODUCT GRAPH OF ۵ሺ‫ , ܖ܈‬૎ሻ WITH ۵ሺ‫ ܖ܄‬ሻ
In this paper we consider the strong product graph of Euler totient Cayley graph with
Arithmetic Vn graph. The properties and some domination parameters of these graphs are studied by
Uma Maheswari [5].
Let ‫ܩ‬ଵ denote the Euler Totient Cayley graph and ‫ܩ‬ଶ denote the Arithmetic Vn graph. Since
‫ܩ‬ଵ and ‫ܩ‬ଶ are two simple graphs, by the definition of adjacency in strong product, ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is a
simple graph.
The strong product graph ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is a complete graph, if ݊ is a prime and the degree of a
vertex in ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is given by
݀݁݃ ீభٝீమ ൫‫ݑ‬௜ , ‫ݒ‬௝ ൯ ൌ ݀݁݃ீభ ሺ‫ݑ‬௜ ሻ ൅݀݁݃ீమ ൫‫ݒ‬௝ ൯ ൅ ݀݁݃ீభ ሺ‫ݑ‬௜ ሻ · ݀݁݃ீమ ൫‫ݒ‬௝ ൯.
= ߮ ሺ݊ሻ ൅ ݀݁݃ீమ ൫‫ݒ‬௝ ൯ ൅ ߮ ሺ݊ሻ. ݀݁݃ீమ ൫‫ݒ‬௝ ൯.
5. STRONG DOMINATING SETS OF STRONG PRODUCT GRAPH
In this section we find minimum strong dominating sets of strong product graph of ‫ܩ‬ሺܼ௡ , ߮ሻ
with ‫ ܩ‬ሺܸ ሻ graph and obtain its strong domination number in various cases.
௡
Strong domination
Let ‫ ܩ‬ሺܸ, ‫ܧ‬ሻ be a graph and ‫ .ܸ א ݒ ,ݑ‬Then, ‫ ݑ‬strongly dominates ‫ ݒ‬if (i) ‫ ܧ א ݒݑ‬and (ii)
݀݁݃ ‫ ݑ‬൒ ݀݁݃ ‫ .ݒ‬A set ‫ ܸ ؿ ܦ‬is called a strong-dominating set of ‫ ܩ‬if every vertex in ܸ െ ‫ ܦ‬is
strongly dominated by at least one vertex in ‫ .ܦ‬The strong domination number ߛ௦ of G is the
minimum cardinality of a strong dominating set.
Some results on strong domination for general graphs can be seen in [6].
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4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
Theorem 5.1: If ݊ ൌ ‫ ,݌‬then the strong domination number of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is 1.
Proof: Let ݊ be a prime. Then the graph ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is a complete graph and hence every vertex is of
degree ݊ െ 1. So, any single vertex set dominates all other vertices in ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
Let ‫ ܦ‬ൌ ሼሺ‫݌ ,ݐ‬ሻሽ, where ‫ ݐ‬is any vertex in ‫ܩ‬ଵ and ‫ ݌‬is a vertex in ‫ܩ‬ଶ .Let ܸ denote the vertex set of
‫ܩ‬ଵ ٝ ‫ܩ‬ଶ . Then every vertex in ܸ െ ‫ ܦ‬is adjacent with the vertex ሺ‫݌ ,ݐ‬ሻ in ‫ .ܦ‬Obviously the degree of
every vertex in ܸ െ ‫ ܦ‬of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is equal to the degree of ሺ‫݌ ,ݐ‬ሻ in ‫ ܦ‬of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ . Thus ‫ ܦ‬ൌ ሼሺ‫݌ ,ݐ‬ሻሽ
is a minimum strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
Therefore ߛ௦ ሺ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ 1.
Theorem 5.2: If ݊ ൌ 2‫ ݌ ,݌‬is an odd prime, then the strong domination number of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is 2.
Proof: Let ݊ ൌ 2‫ ,݌‬where ‫ ݌‬is an odd prime. Consider the graph ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ . Let ܸଵ , ܸଶ and ܸ denote
the vertex sets of ‫ܩ‬ଵ , ‫ܩ‬ଶ and ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ respectively. Then ܸଵ ൌ ሼ0,1,2, … ,2‫ ݌‬െ 1ሽ,
ܸଶ ൌ ሼ2, ‫݌2 ,݌‬ሽ and ܸሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ܸଵ ൈ ܸଶ ൌ ܸ. By Theorem 2.2, the strong domination number of
‫ܩ‬ଵ is 2. Let ‫ܦ‬ଵ ൌ ሼ‫ݑ‬ௗభ , ‫ݑ‬ௗమ ሽ be a strong dominating set of ‫ܩ‬ଵ ,
whereห ‫ݑ‬ௗభ െ ‫ݑ‬ௗమ ห ൌ ‫ . ݌‬Again by Theorem 3.1, ߛ௦ ሺ‫ܩ‬ଶ ሻ ൌ ݇ െ 1 ൌ 2 െ 1 ൌ 1. Clearly ‫ܦ‬ଶ ൌ ሼ2‫݌‬ሽ
is a strong dominating set of ‫ܩ‬ଶ .
Consider ‫ ܦ‬ൌ ‫ܦ‬ଵ ൈ ‫ܦ‬ଶ ൌ ൛൫‫ݑ‬ௗభ , 2‫݌‬൯, ൫‫ݑ‬ௗమ , 2‫݌‬൯ ൟ .We claim that ‫ ܦ‬is a dominating set of
‫ܩ‬ଵ ٝ ‫ܩ‬ଶ . Let ሺ‫ݒ ,ݑ‬ሻ ‫ ܸ א‬െ ‫ .ܦ‬The following cases arise.
Case 1: Suppose ‫ ݑ‬ൌ ‫ݑ‬ௗభ , ‫ ݒ‬ൌ 2 or ‫ .݌‬Then by the definition of strong product, the vertices
൫‫ݑ‬ௗభ , 2൯ and ൫‫ݑ‬ௗభ , ‫݌‬൯ in ܸ െ ‫ ܦ‬are adjacent with ൫‫ݑ‬ௗభ , 2‫݌‬൯, because 2 and ‫ ݌‬are adjacent
with 2‫ ݌‬as ‫ ܦܥܩ‬ሺ2, 2‫݌‬ሻ ൌ 2 and ‫ܦܥܩ‬ሺ‫݌2 ,݌‬ሻ ൌ ‫.݌‬
Case 2: Suppose ‫ ݑ‬ൌ ‫ݑ‬ௗమ , ‫ ݒ‬ൌ 2 or ‫ .݌‬By the similar argument as in Case 1, vertices ൫‫ݑ‬ௗమ , 2൯ and
൫‫ݑ‬ௗమ , ‫݌‬൯ in ܸ െ ‫ ܦ‬are adjacent with ൫‫ݑ‬ௗమ , 2‫݌‬൯.
Case 3: Suppose ‫ݑ ് ݑ‬ௗభ and ‫ݑ ് ݑ‬ௗమ , ‫ ݒ‬ൌ 2 or ‫ ݌‬or 2‫ .݌‬Since ‫ܦ‬ଵ is a dominating set of ‫ܩ‬ଵ , ‫ݑ‬
is adjacent with either ‫ݑ‬ௗభ or ‫ݑ‬ௗమ , say ‫ݑ‬ௗభ . Then by the definition of strong product, vertex ሺ‫ ݒ ,ݑ‬ሻ
is adjacent with the vertex ൫‫ݑ‬ௗభ , 2‫݌‬൯.
Thus ሺ‫ݒ ,ݑ‬ሻ in ܸ െ ‫ ܦ‬is dominated by at least one vertex in ‫ .ܦ‬Therefore ‫ ܦ‬becomes a
dominating set.
Now we show that ‫ ܦ‬ൌ ൛൫‫ݑ‬ௗభ , 2‫݌‬൯, ൫‫ݑ‬ௗమ , 2‫݌‬൯ ൟ is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
We know that ‫ ܩ‬ሺܼ௡ , ߮ሻ is ߮ሺ݊ሻ - regular and for ݊ ൌ 2‫߮ ,݌‬ሺ݊ሻ = ሺ‫ ݌‬െ 1ሻ and so each vertex
has degree ‫ ݌‬െ 1. For ݊ ൌ 2‫ ,݌‬the graph ‫ܩ‬ଶ contains the vertices ሼ2, ‫݌2 ,݌‬ሽ and degሺ2‫݌‬ሻ ൐ deg ሺ2ሻ,
݀eg ሺ‫݌‬ሻ as ‫ܦܥܩ‬ሺ2,2‫݌‬ሻ ൌ 2, ‫ܦܥܩ‬ሺ‫݌2 ,݌‬ሻ ൌ ‫ ݌‬and ‫ܦܥܩ‬ሺ2, ‫݌‬ሻ ൌ 1.
If ൫‫ݑ‬௜ , ‫ݒ‬௝ ൯ is any vertex of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ then we know that
݀݁݃ீభٝீమ ൫‫ݑ‬௜ , ‫ݒ‬௝ ൯ ൌ ݀݁݃ீభ ሺ‫ݑ‬௜ ሻ ൅ ݀݁݃ீమ ൫‫ݒ‬௝ ൯ ൅ ݀݁݃ீభ ሺ‫ݑ‬௜ ሻ · ݀݁݃ீమ ൫‫ݒ‬௝ ൯.
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5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
Then for ݅ ൌ 1 , 2, we have
and
݀݁݃ீభ ٝீమ ሺ‫݀ݑ‬௜ , 2‫݌‬ሻ ൌ ݀݁݃ீభ ሺ‫݀ݑ‬௜ ሻ ൅ ݀݁݃ீమ ሺ2‫݌‬ሻ ൅ ݀݁݃ீభ ሺ‫݀ݑ‬௜ ሻ · ݀݁݃ீమ ሺ2‫݌‬ሻ
ൌ ‫ ݌‬െ 1 ൅ 2 ൅ ሺ‫ ݌‬െ 1ሻ2 ൌ 3‫ ݌‬െ 1,
݀݁݃ீభ ٝீమ ሺ‫݀ݑ‬௜ , 2ሻ ൌ ݀݁݃ீభ ሺ‫݀ݑ‬௜ ሻ ൅ ݀݁݃ீమ ሺ2ሻ ൅ ݀݁݃ீభ ሺ‫݀ݑ‬௜ ሻ · ݀݁݃ீమ ሺ2ሻ
ൌ ‫ ݌‬െ 1 ൅ 1 ൅ ሺ‫ ݌‬െ 1ሻ1 ൌ 2‫ ݌‬െ 1,
݀݁݃ீభ ٝீమ ሺ‫݀ݑ‬௜ , ‫݌‬ሻ ൌ ݀݁݃ீభ ሺ‫݀ݑ‬௜ ሻ ൅ ݀݁݃ீమ ሺ‫݌‬ሻ ൅ ݀݁݃ீభ ሺ‫݀ݑ‬௜ ሻ · ݀݁݃ீమ ሺ‫݌‬ሻ
ൌ ‫ ݌‬െ 1 ൅ 1 ൅ ሺ‫ ݌‬െ 1ሻ1 ൌ 2‫ ݌‬െ 1.
i.e., ݀ሺ‫݀ݑ‬௜ , 2‫݌‬ሻ ൐ ݀ሺ‫݀ݑ‬௜ , 2ሻ and ݀ሺ‫݀ݑ‬௜ , ‫݌‬ሻ--------- (1)
Now consider the vertices in ܸ െ ‫ ,ܦ‬which are
ሼሺ‫݀ݑ‬ଵ , 2ሻ, … , ሺ‫݀ݑ‬௡ , 2ሻ, ሺ‫݀ݑ‬ଵ , ‫݌‬ሻ, … ሺ‫݀ݑ‬௡ , ‫݌‬ሻ, ሺ‫݀ݑ‬ଷ , 2‫݌‬ሻ, … ሺ‫݀ݑ‬௡ , 2‫݌‬ሻሽ.
By (1), it is clear that these vertices are strongly dominated by at least one vertex in ‫ .ܦ‬Therefore ‫ܦ‬
is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
We show that ‫ ܦ‬is minimum. Suppose we delete a vertex, say ൫‫ݑ‬ௗభ , 2‫݌‬൯ from ‫ .ܦ‬Then the
vertices ൫‫ݑ‬ௗభ , 2൯ and ൫‫ݑ‬ௗభ , ‫݌‬൯ are not dominated by the remaining vertex ൫‫ݑ‬ௗమ , 2‫݌‬൯. This is because
‫ݑ‬ௗభ ് ‫ݑ‬ௗమ and ‫ݑ‬ௗభ is not adjacent with ‫ݑ‬ௗమ as ห ‫ݑ‬ௗభ െ ‫ݑ‬ௗమ ห ൌ ‫. ݌‬
Similar is the case if we delete the vertex ൫‫ݑ‬ௗమ , 2‫݌‬൯ from ‫ .ܦ‬Thus ‫ ܦ‬becomes a minimum strong
dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ with cardinality 2.
Therefore ߛ௦ ሺ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ 2.
ఈ
ఈ
ఈ
Theorem 5.3: If ݊ ് ‫ ݌2 ് ݊ ,݌‬and ݊ ൌ ‫݌‬ଵ భ ‫݌‬ଶ మ … ‫݌‬௞ ೖ where ‫݌‬ଵ , ‫݌‬ଶ , … , ‫݌‬௞ are distinct primes
and ߙ௜ ൒ 1, then the strong domination number of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ is given by
ߛ௦ ሺ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ൜
ሺߣ ൅ 1ሻ · ሺ݇ െ 1ሻ
ሺߣ ൅ 1ሻ · ݇
if ߙ௜ ൌ 1 for more than one i,
otherwise.
where ߣ is the length of the longest stretch of consecutive integers in ܸଵ of ‫ܩ‬ଵ each of which shares
a prime factor with n and k is the core of n.
ఈ
ఈ
ఈ
Proof: Let ݊ be neither a prime nor 2‫ .݌‬suppose ݊ = ‫݌‬ଵ భ ‫݌‬ଶ మ … … . . ‫݌‬௞ ೖ , where ߙ௜ ൒ 1.
By Theorem 2.3, the strong domination number of ‫ܩ‬ଵ is ߣ ൅ 1.
Let ‫ܦ‬ଵ ൌ ൛ ‫ݑ‬ௗభ , . . , ‫ݑ‬ௗഊశభ ൟ be a strong dominating set of ‫ܩ‬ଵ , where ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … … . . , ‫ݑ‬ௗഊశభ are
consecutive integers. Again by Theorem 3.1,
݇െ1
if α୧ ൌ 1 for more than one i,
݇
otherwise.
where ݇ is the core of ݊.
Now we have the following possibilities.
ߛ௦ ሺ‫ܩ‬ଶ ሻ ൌ ቄ
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6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
Case 1: Suppose ߙ௜ ൐ 1 for all ݅. By Theorem 3.1, the strong dominating set of ‫ܩ‬ଶ with minimum
cardinality ݇ is given by ‫ܦ‬ଶ ൌ ሼ ‫݌‬ଵ , ‫݌‬ଶ , … . . , ‫݌‬௞ ሽ.
Consider ‫ ܦ‬ൌ ‫ܦ‬ଵ ൈ ‫ܦ‬ଶ ൌ ൛ ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … … . . , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ ‫݌‬ଵ , ‫݌‬ଶ , … . . , ‫݌‬௞ ሽ. We claim that ‫ܦ‬
is a dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
Let ሺ‫ ݒ ,ݑ‬ሻ ‫ ܸ א‬െ ‫ .ܦ‬Then ‫ ݑ‬is adjacent with ‫ݑ‬ௗ೗ for some ݈, where 1 ൑ ݈ ൑ ߣ ൅ 1, because
‫ܦ‬ଵ is a dominating set of ‫ܩ‬ଵ . The vertex ‫ ݒ‬in ‫ܩ‬ଶ is adjacent with any vertex ‫݌‬ଵ , ‫݌‬ଶ , … . . , ‫݌‬௞ say
‫݌‬௠ , as ‫ܦ‬ଶ is a dominating set of ‫ܩ‬ଶ . Hence every vertex ሺ‫ݒ ,ݑ‬ሻ of ܸ െ ‫ ܦ‬is adjacent with at
least one vertex ൫ ‫ݑ‬ௗ೗ , ‫݌‬௠ ൯ in ‫ .ܦ‬Thus ‫ ܦ‬becomes a dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
Now we show that ‫ ܦ‬ൌ ൛ ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … … . . , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ ‫݌‬ଵ , ‫݌‬ଶ , … . . , ‫݌‬௞ ሽ is a strong dominating
set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
For any vertex ሺ‫݀ݑ‬௜ , ‫݌‬௜ ሻ of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ we have
݀݁݃ீభٝீమ ሺ‫݀ݑ‬௜ , ‫݌‬௜ ሻ ൌ ߮ሺ݊ሻ ൅ ݀݁݃ீమ ሺ‫݌‬௜ ሻ ൅ ߮ሺ݊ሻ · ݀݁݃ீమ ሺ‫݌‬௜ ሻ,
൒ ߮ሺ݊ሻ ൅ ݀݁݃ீమ ൫‫ݒ‬௝ ൯ ൅ ߮ሺ݊ሻ · ݀݁݃ீమ ൫‫ݒ‬௝ ൯, where ‫݌‬௜ ് ‫ݒ‬௝ ,
൒ ݀݁݃ீభ ٝீమ ൫‫݀ݑ‬௜ , ‫ݒ‬௝ ൯,
as ‫݌‬௜ , ݅ ൌ 1,2,3, … , ݇ has maximum degree in ‫ܩ‬ଶ , ݅ ൌ 1,2,3, … , ݇.
That is ݀݁݃ீభٝீమ ሺ‫݀ݑ‬௜ , ‫݌‬௜ ሻ ൒ the degree of all other vertices in ܸ െ ‫ .ܦ‬So it is clear that every
vertex in ܸ െ ‫ ܦ‬is strongly dominated by at least one vertex in ‫.ܦ‬
Therefore ‫ ܦ‬is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
We now show that ‫ ܦ‬is minimum. Suppose we delete a vertex ൫ ‫ݑ‬ௗ೔ , ‫݌‬௝ ൯ from ‫.ܦ‬
Let the vertex ‫ݑ‬ௗ೔ be adjacent with the vertices ‫ݑ‬ଵ , ‫ݑ‬ଶ , ‫ݑ‬ଷ ,…., ‫ݑ‬ఝሺ௡ሻ as degree of each vertex in
‫ܩ‬ଵ is ߮ሺ݊). Here all the vertices ‫ݑ‬ଵ , ‫ݑ‬ଶ , ‫ݑ‬ଷ ,…., ‫ݑ‬ఝሺ௡ሻ are not adjacent with the vertices of ‫ܦ‬ଵ െ
൛ ‫ݑ‬ௗ೔ ൟ. If this is happens, then ‫ܦ‬ଵ െ ൛ ‫ݑ‬ௗ೔ ൟ becomes a dominating set of ‫ܩ‬ଵ , a contradiction to the
minimality of ‫ܦ‬ଵ .
Therefore there is at least one vertex in ൛‫ݑ‬ଵ , ‫ݑ‬ଶ , ‫ݑ‬ଷ , … . , ‫ݑ‬ఝሺ௡ሻ ൟ , say ‫ݑ‬௥ , which is not
adjacent with any vertex of ‫ܦ‬ଵ െ ൛ ‫ݑ‬ௗ೔ ൟ.
Since ‫݌‬ଵ , ‫݌‬ଶ , … . . , ‫݌‬௞ are distinct primes, there is no adjacency between these vertices. Then
by the definition of strong product, vertex ൫ ‫ݑ‬௥ , ‫݌‬௝ ൯ is not adjacent with any vertex of ‫ ܦ‬െ
൛൫ ‫ݑ‬ௗ೔ , ‫݌‬௝ ൯ ൟ. Hence ‫ ܦ‬െ ൛൫ ‫ݑ‬ௗ೔ , ‫݌‬௝ ൯ ൟ is not a dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
Thus ‫ ܦ‬becomes a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ with minimum cardinality ሺߣ ൅ 1ሻ. k.
Hence ߛ௦ ሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ሺߣ ൅ 1ሻ. k.
ఈ
ఈ
ఈ
ఈ
ఈ
೔షభ
೔శభ
Case 2: Suppose ߙ௜ ൌ 1, for only one ݅. That is ݊ = ‫݌‬ଵ భ ‫݌‬ଶ మ … … ‫݌‬௜ିଵ ‫݌‬௜ ‫݌‬௜ାଵ … . . ‫݌‬௞ ೖ .
By Theorem 3.1, the strong dominating set of ‫ܩ‬ଶ is given by
൛ ‫݌‬ଵ , . . ‫݌‬௜ିଵ , ‫݌‬௜ ‫݌‬௝ , ‫݌‬௜ାଵ , . . , ‫݌‬௞ ൟ and deg ሺ ‫݌‬௜ ‫݌‬௝ ሻ ൐ deg ሺ‫݌‬௜ ሻ.
Let ‫ ܦ‬ൌ ൛ ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … … . . , ‫ݑ‬ௗഊశభ ൟ ൈ ൛ ‫݌‬ଵ , ‫݌‬ଶ , … … ‫݌‬௜ିଵ , ‫݌‬௜ ‫݌‬௝ , ‫݌‬௜ାଵ , … . , ‫݌‬௞ ൟ.
In a similar way to Case 1, we can prove that ‫ ܦ‬is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ with
minimum cardinality ሺߣ ൅ 1ሻ. ݇.
Therefore ߛ௦ ሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ሺߣ ൅ 1ሻ. k.
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7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
Case 3: Let ߙ௜ ൌ 1 for all ݅. That is ݊ ൌ ‫݌‬ଵ ‫݌‬ଶ … … . ‫݌‬௞ .
Sub Case (i): Suppose ݇ ൌ 2. Then ݊ ൌ ‫݌‬ଵ . ‫݌‬ଶ .
Let ‫ ܦ‬ൌ ‫ܦ‬ଵ ൈ ‫ܦ‬ଶ ൌ ൛ ‫ݑ‬ௗభ , … . . , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ‫݌‬ଵ ‫݌‬ଶ ሽ.
In a similar way to Theorem 5.2, we can show that ‫ ܦ‬is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ
with minimum cardinality ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
Therefore ߛ௦ ሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
Sub Case (ii): Suppose ݇ ൌ 3. As in Theorem 3.1, we can see that the strong dominating
set of ‫ܩ‬ଶ is given by ሼ‫݌‬ଵ , ‫݌‬ଶ ‫݌‬ଷ ሽ. Here we observe that the vertex ‫݌‬ଵ ‫݌‬ଶ ‫݌‬ଷ is adjacent with the
vertices ‫݌‬ଵ , ‫݌‬ଶ , ‫݌‬ଷ only and hence we have to include either one of these vertices into the dominating
set of ‫ܩ‬ଶ in order to dominate the vertex ‫݌‬ଵ ‫݌‬ଶ ‫݌‬ଷ .
Let ‫ ܦ‬ൌ ‫ܦ‬ଵ ൈ ‫ܦ‬ଶ ൌ ൛ ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … . . , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ‫݌‬ଵ , ‫݌‬ଶ ‫݌‬ଷ ሽ. Then we can show as in Theorem
5.2 that ‫ ܦ‬is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ with minimum cardinality
ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
Hence ߛ௦ ሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ |‫ |ܦ‬ൌ ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
Sub Case (iii): Suppose݇ ൐ 3. As in Theorem 3.1 we can see that the strong dominating
set of ‫ܩ‬ଶ is given by ሼ‫݌‬ଵ , ‫݌‬ଵ ‫݌‬ଶ , ‫݌‬ଶ ‫݌‬ଷ , … , ‫݌‬௞ିଷ ‫݌‬௞ିଶ , ‫݌‬௞ିଵ ‫݌‬௞ ሽ and ݀݁݃൫‫݌‬௜ ‫݌‬௝ ൯ ൐ ݀݁݃൫‫݌‬௜ ‫݌‬௝ ‫݌‬௟ ൯ ൐ ‫ ڮ‬൐
݀݁݃ሺ‫݌‬ଵ ‫݌‬ଶ ‫݌‬ଷ … . ‫݌‬௞ ሻ and ݀݁݃ሺ‫݌‬௜ ሻ ൏ ݀݁݃൫‫݌‬௜ ‫݌‬௝ ൯; degሺ‫݌‬௜ ሻ ൐ deg൫‫݌‬௜ ‫݌‬௝ ‫݌‬௟ ൯,………..,
൐ degሺ‫݌‬ଵ ‫݌‬ଶ ‫݌‬ଷ … ‫݌‬௞ ሻ , ݅ ൌ 1,2, … , ݇.
Let ‫ ܦ‬ൌ ‫ܦ‬ଵ ൈ ‫ܦ‬ଶ ൌ ൛ ‫ݑ‬ௗభ , … , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ‫݌‬ଵ , ‫݌‬ଵ ‫݌‬ଶ , … , ‫݌‬௞ିଷ ‫݌‬௞ିଶ , ‫݌‬௞ିଵ ‫݌‬௞ ሽ.
As per the explanation given in Sub case (ii) and again in a similar way to Theorem 5.2 we
can show that ‫ ܦ‬is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ with minimum cardinality
ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
Therefore ߛ௦ ሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
Case 4: Suppose α୧ ൌ 1 for some i.
ఈ
ఈ
Sub case (i): Suppose αଵ ൌ 1, αଶ ൌ 1. That is ݊ ൌ ‫݌‬ଵ . ‫݌‬ଶ . ‫݌‬ଷ య . … . . ‫݌‬௞ ೖ .
By Theorem 3.1, the strong dominating set of ‫ܩ‬ଶ is given by
ሼ‫݌‬ଷ , ‫݌‬ସ , … . . , ‫݌‬௞ , ‫݌‬ଵ ‫݌‬ଶ ሽ and ݀݁݃ሺ‫݌‬ଵ ‫݌‬ଶ ሻ ൐ ݀݁݃ሺ‫݌‬ଵ ሻ, ݀݁݃ሺ‫݌‬ଶ ሻ.
Let ‫ ܦ‬ൌ ‫ܦ‬ଵ ൈ ‫ܦ‬ଶ ൌ ൛ ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … . , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ‫݌‬ଷ , ‫݌‬ସ , … . . , ‫݌‬௞ , ‫݌‬ଵ ‫݌‬ଶ ሽ.
In a similar way to Case 1, we can prove that ‫ ܦ‬is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ with
minimum cardinality ሺߣ ൅ 1ሻ. ሺ݇ െ 1ሻ.
Therefore ߛ௦ ሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
ఈ
ఈ
Sub case (ii): Suppose αଵ ൌ αଶ ൌ αଷ ൌ 1. That is ݊ ൌ ‫݌‬ଵ . ‫݌‬ଶ . ‫݌‬ଷ ‫݌‬ସ ర . … . . ‫݌‬௞ ೖ .
Let ‫ ܦ‬ൌ ‫ܦ‬ଵ ൈ ‫ܦ‬ଶ ൌ ൛ ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … . , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ‫݌‬ସ , ‫݌‬ହ , … . , ‫݌‬௞ , ‫݌‬ଵ , ‫݌‬ଶ ‫݌‬ଷ ሽ.
Then we can show as in above Sub case (i), that ‫ ܦ‬is a strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ with
cardinality ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
Sub case (iii): Suppose α୨ ൌ 1 for ݆ ൌ 1,2, … . . , ݅ and α୨ ൐ 1 for ݆ ൌ ݅ ൅ 1, ݅ ൅ 2, … . . , ݇.
ఈ೔శభ
ఈ
Then ݊ ൌ ‫݌‬ଵ . ‫݌‬ଶ . ‫݌‬ଷ … … ‫݌‬௜ . ‫݌‬௜ାଵ . … . . ‫݌‬௞ ೖ .
We can prove as in Case 1 and Sub case (iii) of Case 3 that the set
‫ ܦ‬ൌ ൛ ‫ݑ‬ௗభ , ‫ݑ‬ௗమ , … , ‫ݑ‬ௗഊశభ ൟ ൈ ሼ‫݌‬௜ାଵ , ‫݌‬௜ାଶ , . . , ‫݌‬௞ , ‫݌‬ଵ , ‫݌‬ଵ ‫݌‬ଶ , … , ‫݌‬௜ିଷ ‫݌‬௜ିଶ , ‫݌‬௜ିଵ ‫݌‬௜ ሽ forms a minimum
strong dominating set of ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ .
Therefore ߛ௦ ሺ ‫ܩ‬ଵ ٝ ‫ܩ‬ଶ ሻ ൌ ሺߣ ൅ 1ሻ. ሺk െ 1ሻ.
142

8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
ILLUSTRATIONS
࢔ ൌ ૚૚
ࡳ૛ ൌ ࡳሺࢂ૚૚ ሻ
ࡳ૚ ൌ ࡳሺࢆ૚૚ , ࣐ሻ
ࡳ ૚ ٝ ࡳ૛
Strong dominating set = {(0,11)}
࢔ൌ૛ൈ૜ൌ૟
ࡳ૚ ൌ ࡳሺࢆ૟ , ࣐ሻ
ࡳ૛ ൌ ࡳሺ ࢂ ૟ ሻ
ࡳ ૚ ٝ ࡳ૛
Strong dominating set = {(0,6), (3,6)}
143

9. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
࢔ ൌ ૛૛ ൌ ૝
ࡳ૚ ൌ ࡳሺࢆ૝ , ࣐ሻ
ࡳ૛ ൌ ࡳሺ ࢂ ૝ ሻ
ࡳ ૚ ٝ ࡳ૛
Strong dominating set = {(0,2), (1,2)}
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