The vulnerability of communication network measures the resistance of network to the disruption of operation after the failure of certain station or communication links. For any communication network greater degrees of stability or less vulnerability is required. Vulnerability can be measured by certain parameters like connectivity, toughness, integrity, binding number etc. In the analysis of vulnerability of communication network to disruption, following two parameters are of great importance: 1) The size of the largest remaining group within which mutual communication can still occur; 2) The number of elements that are not functioning. In this context Barefoot et al. [
Definition 1.1. The integrity of a graph G is denoted by 
and defined by 
where m(G – S) is the order of a maximum component of 
Definition 1.2. An I-set of G is any (proper) subset S of 
for which 
The connectedness of graph is not essential to define integrity. The integrity of middle graphs is discussed by Mamut and Vumar [
Definition 1.3. The domination integrity of a connected graph G is denoted as 
and defined by 
where 
is the order of a maximum component of
.
Sundareswarn and Swaminathan [
Definition 1.4. Duplication of a vertex 
by a new edge 
in graph G produces a new graph 
such that 
and
.
Definition 1.5. Duplication of an edge 
by a new vertex 
in a graph G produces a new graph 
such that 
Definition 1.6. For the dominating set 
a vertex 
is called isolate of 
if 
For all other standard terminology and graph theoretical notation we refer to Hynes et al. [
then 
of path 
by vertex 
There are three types of vertices in G
for 
and
must be in the dominating set as they are the most dominating vertices. From the nature of the graph G it is obvious that out of the vertices 
and 
at least one vertex must belongs to any dominating set S as 
is adjacent to only 
and 
Therefore if S is any dominating set then 
when n is even and 
when 
is odd are minimal dominating sets.
and 
are adjacent to v2i and removal of 
from set S, 
will not be dominated by any vertex of S Hence 
then
: Let G be a graph obtained by duplication of an edge 
of path 
by a vertex 
Then 
and 
as a γ-set of G and then 
This implies 
If 
is any dominating set with 
then 
and consequently 
Hence in all the cases 
: Let G be a graph obtained by duplication of an edge v1v2 and v2v3 of path 
by the vertices u1 and u3 respectively. As 
is the only γ-set of G then 
and m(G – D) = 2. If S1 is any dominating set with 
then 
and
. If S2 is any dominating set with 
then 
and 
Moreover 
is not possible, as the order of the largest component of 
is at most 3. Thus
.
: Let G be a graph obtained by duplication of an edge 
of path 
by vertex 
Then 
with 
is a γ- set of 
and 
Consequently
If 
is any dominating set with
then 
and
If 
is any dominating set with 
then 
and 
Hence we have
.
be a graph obtained by duplication of each edge by vertex of path 
then
(if
)
where 
If n is even then 
is a γ-set of G otherwise 
is a γ-set of G Therefore 
which implies,
is minimum. For that we have to take into account the minimality of both 
and
. The minimality of 
is guaranteed as S is γ-set. Now it remains to show that if S is any dominating set other than D then 
then 
then 
then trivially
Hence for any dominating set S
(if n ≥ 5).
be a graph obtained by duplication of each vertex by an edge of path 
or cycle 
then 
be a graph obtained by duplication of vertices 
of path 
or cycle 
by an edge 
Then from the construction of graph G it is obvious that from the vertices 
and 
at least one vertex must belong to any dominating set 
Consequently if 
is any dominating set then 
is a minimal dominating set. Because each 
is adjacent to 
and 
If 
is removed from set 
then 
and 
will not be dominated by any vertex. Thus 
is a minimal dominating set with minimum cardinality. Hence 
be a graph obtained by duplication of each vertex of path 
or cycle 
by an edge then 
be a graph obtained by duplication of each vertex 
of path 
or cycle 
by an edge 
Then from Theorem 2.4 we have 
Let 
be a γ-set of graph G. Then 
Therefore,
is minimum. For that we have to take into account the minimality of both 
and 
The minimality of 
is guaranteed as S is γ-set. Now it remains to show that if S is any dominating set other than D then 
. If 
then
, consequently 
If 
then trivially 
Hence for any dominating set S,
is a minimal dominating set if and only if for each vertex
, one of the following two conditions holds:
.
for which 
by an edge then 
be a graph obtained by duplication of rim vertices as well as apex vertex altogether of wheel 
by edges
and 
respectively. Then each rim vertex 
will dominate the vertices 
and apex vertex 
For 
there exists a vertex 
such that 
is a singleton set. Then from Proposition 2.6 
will be a minimal dominating set of 
If 
is any dominating set then we claim that 
Because
are only of the type 
then 
are combination of 
and 
then 
contains any of first two types together with the apex vertex then 
contains 
and apex vertex then 
be a graph obtained by duplication of each vertex of wheel 
by an edge then 
be a graph obtained by duplication of apex vertex 
of wheel 
by an edge 
and the rim vertices 
of wheel 
by an edge 
Then from Lemma 2.7 we have 
Let 
be a γ-set of graph G. Then 
Therefore
is minimum. For that we have to take into account the minimality of both 
and 
The minimality of 
is guaranteed as S is γ-set. Now it remains to show that if S is any dominating set other than D then
then
consequently
then 
consequently
then trivially 
Thus for any dominating set S,