Design of interconnection networks is an important integral part of any parallel processing of distributed system. There has been a strong interest recently in using Cayley graphs as a model for developing interconnection networks for large interacting arrays of CPU’s. An excellent survey of interconnection networks based on Cayley graphs can be found in [
Let Γ be a finite group. Let Ω be a generating set of Γ satisfying e Ï Ω and a Î Ω implies a−1 Î Ω. The Cayley graph corresponding to Γ is the graph G = (V, E), where V(G) = Γ and E(G)={(x, xa): x Î V(G), a Î Ω} and it is denoted by G = Cay(Γ, Ω). Let G= (V, E), be a finite, simple and undirected graph. We follow the terminology of [
An E-chain is a countable family of nested graphs, each of which has an E-set. We say that a countable family of graphs G = {Gi, i ³ 1} with each Gi has an E-set Si is an inclusive E-chain if for every i ³ 1, there exists a surjective map fi: Gi+1 ® Gi such that 
(Si) Ì Si+1. And also we define that a finite family of graphs G = {Gi, i ³ 0} is an inductive E-chain if every Gi+1 is a spanning subgraph of Gi and each Gi has an E-set Si. Let V(Gi) be any finite group and if, for each i ³ 0, there exists a bijective map zi: V(Gi) ® V(Gi+1) such that zi(Si ) Í Si+1 and Si is the subgroup of V(Gi) then we say that G is an inductive subgroups E-chain.
A graph 
is called a covering of G with projection 
if there is a surjection 
such that 
is a bijection for any vertex 
and
. We use the covering function to show the inclusive E-chain.
In this paper, we obtain upper bounds for domination number, total domination number and connected domination number in a Cayley graph 
constructed on the dihedral group
, for 
and a generating set
. Further, we obtain some E-sets in
. Note that the dihedral group 
with identity e is the group generated by two elements r and s with 
and
. From these defining relations, one can take
and
, where 
is a generating set of
. Throughout this paper, 
be an integer, 
,
and k, t be integers such that 
. We take the generating set 
in the form that
where 
and 
Let 
for
, 
for 
and
. Some of the results are listed below for further reference.
Theorem 1 [
, with the equality if and only if G has an efficient dominating set.
Theorem 2 [
be a covering and let S be a perfect domination set of G. Then 
is a perfect domination set of
. Moreover, if S is independent, then 
is independent.
Theorem 3 [
is cyclic or dihedral. A complete listing of the subgroups is as follows:
1) cyclic subgroups
, where d divides n, with index 2d.
2) dihedral subgroups
, where d divides n and 
with index d. Every subgroup of 
occurs exactly once in this listing.
. Also whenever the equality occurs we give the corresponding sets.
be an integer, 
and kt are integers such that
. Let
. If 
for 
, 
for 
and 
, then
.
and
. Consider the set
and
where 
and 
We have to prove that
. If
, then we can write 
as either one vertex of the form 
or
, where
. By the division algorithm, 
, where 
and
.
. We have the following cases:
and
.
, then by the definition of
.
, for some integers m, g with 
and 
then 
whereas 
and so
.
and 
. In this case, there exists an integer h with 
such that 
then 
, for some integers m, g with 
and
. In this case, 
, which means that 
and
.
such that
.
, then 
.
, for some integers m, g, with 
and
. In this case, 
, which means that 
.
and
.
such that
.
, then
.
, for some integers m, g with 
and
. In this case, 
which means that
.
. We have the following cases:
and 
. In this case, there exists an integer h with 
such that 
, then
.
, for some integers m, g with 
and
. In this case, 
, which means that
.
and 
. In this case, there exists an integer h with 
such that 
.
then 
.
, for some integers m, g with 
and
. In this case, 
, which means that 
.
and 
. In this case, there exists an integer h with 
such that
.
, then by the definition of 
.
, for some integers m, g with 
and
. In this case, 
, which means that 
.
.
be an integer, 
and k, t be integers such that
. Let
. If 
for
, 
for 
and
, then
.
and
. Consider the set
.
. We have to prove that 
If
, then we can write 
as either one vertex of the form 
or
, where
. By the division algorithm, 
, where 
and
. We have the following cases:
and
. For some integer g with 
and by the definition of d, if
, then 
or if
, then
.
and
. We can write
, for some integers m, g with 
and
. If
, then 
whereas 
and so 
or if
, then
whereas 
and so
.
and
. In this case, there exists an integer h with 
such that 
or
.
and if
, then 
or if
then
.
, for some integers m, g with 
and
. In this case, if
, then
, which means that
or if
, then
, which implies that 
.
and
. Then there exists an integer h with 
such that 
or
.
, and if
, then 
or if
, then
.
, for some integers m, g with 
and
. In this case, if 
and
, which means that
or if
, then
, which means that 
.
is a total dominating set of G. 
.
be an integer, 
and k, t be integers such that
. Let
. If 
for
, 
for 
and
, then
.
and
. Consider the set
.
and 
is a total dominating set. Since 
and for each 
with
, we have paths
and 
. Also note that 
and
, 
and 
are connected. Hence the induced subgraph 
is connected.
. Moreover we have identified certain subgroups of 
which are also efficient domination sets in
.
be an integer, 
and k, t be integers such that 
and d is an integer such that 
divides n. Let
. Then
. In this case, 
has an E-set.
and
. In the notation of Lemma 4,
’s and
’s are same, 
for all 
and 
for all
.
and
. By Lemma 4,
. Since
is 
regular, by Theorem 1, one can conclude that 
is an E-set in
.
. Now we us identify some of the subgroups as efficient dominating sets.
be an integer, 
and k, t be integers such that
, 
and 
divides
. Let 
be a subgroup of the dihedral group
, where 
and
, 
Then, there exists a generating set 
of 
such that H is an efficient dominating set for the Cayley graph
.
and
. By taking 
in Theorem 7,
.
is an efficient dominating set for the Cayley graph 
for all
.
. We use this tool to obtain an inclusive E-chain and inductive subgroups E-chain of Cayley graphs on the dihedral group.
be an integer, 
, 
,
Assume that 
divides 
and 
divides
. Then the finite family of graphs 
is inductive subgroups E-chain.
. By the assumption
. divides
. Define the map 
by 
for all
. By Theorem 9, 
has an efficient dominating set and it is of the form
’s are subgroups. It implies that 
for every
. Hence the family of graphs 
is inductive subgroups E-chain.
be an integer, 
, k, t be integers such that
, 
and d is an integer such that 
divides
. For
, let
. Then 
is a covering of
.
by 
and
for all j, where
. Note that 
is a group homomorphism from 
onto
. Let
. Suppose 
and 
are adjacent in
. Then, there exists 
with
or 
with 
such that
or
. Since 
is a group homomorphism and
, we have 
or 
and so 
and 
are adjacent in
. Consider the map 
for any vertex 
and
. Claim 
is bijection. Any element 
in 
as either one vertex of the form 
or
, where
. Let 
Then we have following three cases:
and 
with
. Suppose
, i.e. 
. i.e. 
which is a contradiction to
. Therefore
.
and
. Suppose
, i.e. 
This means
or
, which is a contradiction. Therefore
.
and 
with
. Suppose
, i.e.
. i.e.
which is a contradiction. Therefore 
. Hence distinct elements of 
are distinctly mapped onto 
and so 
is a required bijection.
be an integer, 
k, t, be integers such that
, 
and d is an integer such that 
divides
. For 
let
. Let 
be an efficient dominating set for
. Then the finite family of graphs 
is an inclusive E-chain.
is a covering of
. Since by Theorem 2,
. Hence the finite family of graphs 
is an inclusive E-chain.