# Domination Integrity of Splitting Graph of Path and Cycle

Authors: Samir K. Vaidya; Nirang J. Kothari
Journal: ISRN Combinatorics
 (2013)

---

## Abstract

IfS is a dominating set of a connected graph G then the domination integrity is
the minimum of the sum of two parameters, the number of elements in S and the order
of the maximum component of G−S. We investigate domination integrity of splitting
graph of path Pn and cycle Cn. This work is an effort to relate network expansion and
vulnerability parameter.


---

## Body

## 1. Introduction

The stability of a communication network is of prime importance for any network designer. A shrinking network eventually loses links or nodes, its effectiveness is continuously decreasing and network becomes vulnerable. Many graph theoretic parameters have been introduced for the measurement of vulnerability. Some of them are connectivity, toughness, integrity, and rupture degree. The integrity of a graph is one of the well explored concepts which was introduced by Barefoot et al. [1].Definition 1.
The integrity of a graphG is denoted by I(G) and defined by I(G) = min{|S|+m(G-S):S⊂V(G)}, where m(G-S) is the order of a maximum component of G-S.Definition 2.
AnI-set of G is any (proper) subset S of V(G) for which I(G)=|S|+m(G-S).The integrity of the complete graphKn, path Pn, cycle Cn, star K1,n, complete bipartite graph Km,n, and power graph of cycle were discussed by Barefoot et al. [1, 2] while Goddard and Swart [3, 4] have investigated the bounds for integrity of graphs and its complement. They have also investigated the integrity of graphs in the context of some graph operations. The integrity of middle graphs is discussed by Mamut and Vumar [5] while integrity of total graphs is discussed by Dündar and Aytaç [6].Definition 3.
A setS⊆V of vertices in a graph G=(V,E) is called dominating set if every vertex v∈V is either an element of S or is adjacent to an element of S.Definition 4.
Thedomination numberγ(G) of a graph G equals the minimum cardinality of minimal dominating set of graph G.IfD is any minimal dominating set and if the order of the largest component of G-D is small, then the removal of D will crash the communication network. Considering this aspect, the concept of domination integrity was introduced by Sundareswaran and Swaminathan [7].Definition 5.
Thedomination integrity of a connected graph G is denoted as DI(G) = min{|S|+m(G-S):S is a dominating set}, where m(G-S) is the order of a maximum component of G-S.Sundareswaran and Swaminathan [8] have investigated domination integrity of middle graph of some graphs while Vaidya and Kothari [9] have discussed domination integrity of a graph obtained by duplication of an edge by a vertex and duplication of vertex by an edge in Cn and Pn. In the present work we investigate domination integrity of splitting graphs of path and cycle.Definition 6.
For a graphG the splitting graph S′(G) of graph G is obtained by adding a new vertex v′ corresponding to each vertex v of G such that N(v)=N(v′) where N(v) and N(v′) are the neighborhood sets of v and v′, respectively.For standard graph theoretic notation we refer to West [10] while for terminology related to domination in Graphs we refer to Haynes et al. [11].


## 2. Main Results

Theorem 7.
For alln≥2,
(1)γ(S′(Pn))={n2ifn≡0(mod4)n+12ifn≡1,3(mod4)n+22ifn≡2(mod4).Proof.
Letv1,v2,…,vn be the vertices of path Pn and u1,u2,…,un be the vertices corresponding to v1,v2,…,vn which are added to obtain S′(Pn). As N(u1)={v2} and N(un)={vn-1}, at least one vertex from each pair from u1,v2 and un,vn-1 must belong to any dominating set of S′(Pn). Also at least one vertex from vi-1 and vi+1 must belong to any dominating set as N(ui)={vi-1,vi+1} and N(vi)={vi-1,vi+1,ui-1,ui+1}, where 2≤i≤n-1. Consequently
(2)|S|≥n2foranydominatingsetS.
Now depending upon the number of vertices inPn we consider following subsets for 0≤j<⌊n/4⌋:
(3)S={v2+4j,v3+4j},|S|=n2forn≡0(mod4),S={v2+4j,v3+4j,vn-1},|S|=n+12forn≡1(mod4),S={v2+4j,v3+4j,vn-2,vn-1},|S|=n+22forn≡2(mod4),S={v2+4j,v3+4j,vn-2,vn-1},|S|=n+12forn≡3(mod4).
We claim that each S is dominating set because N(v2+4j)={v1+4j,v3+4j,u1+4j,u3+4j} and N(v3+4j)={v2+4j,v4+4j,u2+4j,u4+4j}.
Also eachS is minimal because the vertex u1+4j will not be dominated by any of the vertices when the vertex v2+4j is removed.
Thus above definedS  is  minimal  dominating  set  of  S′(Pn), hence from (2) we get
(4)γ(S′(Pn))={n2ifn≡0(mod4)n+12ifn≡1,3(mod4)n+22ifn≡2(mod4).Observation 1.
IfS is any dominating set of S′(Pn) with m(S′(Pn)-S)=2 then |S|≥n-2.Observation 2.
IfS is any dominating set of S′(Pn) with m(S′(Pn)-S)=1 then |S|≥n.Theorem 8.
(5)
D
I
(
S
′
(
P
n
)
)
=
{
3
if
n
=
2
n
if
3
⩽
n
⩽
9
n
-
1
if
10
⩽
n
⩽
12
n
-
2
if
13
⩽
n
⩽
15
n
-
3
if
16
⩽
n
⩽
18
15
if
n
=
19
.Proof.
Letv1,v2,…,vn be the vertices of path Pn and u1,u2,…,un be the vertices corresponding to v1,v2,…,vn which are added to obtain S′(Pn).
Case
1
(
n
=
2
). From Theorem 7  γ(S′(P2))=2 and D={v1,v2} is a γ-set of S′(P2). Then m(S′(P2)-D)=1. This implies that DI(S′(P2))=γ(S′(P2))+m(S′(P2)-D)=2+1=3 since γ(S′(P2))≤|S| and m(S′(P2)-D)≤m(S′(P2)-S) for any dominating set S of S′(P2). Consequently γ(S′(P2))+m(S′(P2)-D)≤|S|+m(S′(P2)-S) for any dominating set S of S′(P2). Hence DI(S′(P2))=3.
Case
2
(
3
≤
n
≤
9
). Consider the following subcases.
Subcase (i) (n=3). From Theorem 7  γ(S′(P3))=2 and D={v2,u2} is a γ-set of S′(P3). Then m(S′(P3)-D)=1. This implies that DI(S′(P3))=γ(S′(P3))+m(S′(P3)-D)=2+1=3 since γ(S′(P3))≤|S| and m(S′(P3)-D)≤m(S′(P3)-S) for any dominating set S of S′(P3). Consequently γ(S′(P3))+m(S′(P3)-D)≤|S|+m(S′(P3)-S) for any dominating set S of S′(P3). Hence DI(S′(P3))=3.
Subcase (ii) (4≤n≤6). From Theorem 7  γ(S′(Pn))=n-2 and D={v2,v3,…,vn-1} is a γ-set of S′(Pn). Then m(S′(Pn)-D)=2.
Therefore(6)DI(S′(Pn))≤γ(S′(Pn))+m(S′(Pn)-D)=n-2+2=n.
If S is any dominating set other than D of S′(Pn) with m(S′(Pn)-S)=1 then |S|≥n. This implies that
(7)|S|+m(S′(Pn)-S)≥n+1>n.
Hence from (6) and (7) DI(S′(Pn))=n.
Subcase (iii) (n=7). From Theorem 7  γ(S′(P7))=4 and D={v2,v3,v5,v6} is a γ-set of S′(P7). Then m(S′(P7)-D)=3.
Therefore(8)DI(S′(P7))≤γ(S′(P7))+m(S′(P7)-D)=4+3=7.
Now if S1 is any dominating set other than D of S′(P7) and m(S′(P7)-S1)=2 then |S1|≥5. This implies that
(9)|S1|+m(S′(P7)-S1)≥5+2=7.
Also if S2 is any dominating set other than D and S1 of S′(P7) with m(S′(P7)-S2)=1 then |S2|≥7.
This implies that(10)|S2|+m(S′(P7)-S2)≥7+1=8.
Hence from (8), (9), and (10) DI(S′(P7))=7.
Case
3
(
n
=
10
). From Theorem 7  γ(S′(P10))=6 and D={v2,v3,v5,v6,v8,v9} is a γ-set of S′(P10). Then m(S′(P10)-D)=3.
Therefore(11)DI(S′(P10))≤γ(S′(P10))+m(S′(P10)-D)=6+3=9.
Now if S1 is any dominating set other than D of S′(P10) and m(S′(P10)-S1)=2 then |S1|≥8.
This implies that(12)|S1|+m(S′(P10)-S1)≥8+2=10.
Also if S2 is any dominating set other than D and S1 of S′(P10) with m(S′(P10)-S2)=1 then |S2|≥10.
This implies that(13)|S2|+m(S′(P10)-S2)≥10+1=11.
Hence from (11), (12) and (13) DI(S′(P10))=9.
Case
4
(
n
=
8,9
and
11
≤
n
≤
19
). We know that
(14)DI(S′(Pn))≤γ(S′(Pn))+m(S′(Pn)-D).
The domination number with γ(S′(Cn))+m(S′(Pn)-D) and set S with |S|+m(S′(Pn)-S) are shown in Table 1.
From Table1(15)|S|+m(S′(Pn)-S)≤γ(S′(Pn))+m(S′(Pn)-D).
If S1 is any dominating set of S′(Pn) other than D and S with m(S′(Pn)-S1)=4 or 5 then |S|≤|S1|.
This implies that(16)|S|+m(S′(Pn)-S)≤|S1|+m(S′(Pn)-S1).
If S2 is any dominating set other than D, S1, and S with m(S′(Pn)-S2)=2 then |S2|≥n-2.
This implies that(17)|S2|+m(S′(Pn)-S2)≥(n-2)+2=n.
If S3 is any dominating set other than D, S1, S2, and S with m(S′(Pn)-S3)=1 then |S3|≥n.
This implies that(18)|S3|+m(S′(Pn)-S3)≥n+1.
Hence from Table 1 and the results (14) to (18) we conclude that
(19)DI(S′(Pn))=|S|+m(S′(Pn)-S)={3ifn=2nif3⩽n⩽9n-1if10⩽n⩽12n-2if13⩽n⩽15n-3if16⩽n⩽1815ifn=19.
Hence the result.Table 1
n
γ
(
S
′
(
P
n
)
)
m
(
S
′
(
P
n
)
-
D
)
γ
(
S
′
(
P
n
)
)
+
m
(
S
′
(
P
n
)
-
D
)
S
m
(
S
′
(
P
n
)
-
S
)
|
S
|
+
m
(
S
′
(
P
n
)
-
S
)
8
4
6
10
{
v
2
,
v
3
,
v
4
,
v
6
,
v
7
}
3
8
9
5
6
11
{
v
2
,
v
3
,
v
5
,
v
6
,
v
7
,
v
8
}
3
9
11
6
6
12
{
v
2
,
v
3
,
v
5
,
v
6
,
v
7
,
v
9
,
v
10
}
3
10
12
6
6
12
{
v
2
,
v
3
,
v
5
,
v
6
,
v
7
,
v
8
,
v
10
,
v
11
}
3
11
13
7
6
13
{
v
2
,
v
3
,
v
5
,
v
6
,
v
8
,
v
9
,
v
11
,
v
12
}
3
11
14
8
6
14
{
v
2
,
v
3
,
v
4
,
v
6
,
v
7
,
v
9
,
v
10
,
v
12
,
v
13
}
3
12
15
8
6
14
{
v
2
,
v
3
,
v
4
,
v
6
,
v
7
,
v
8
,
v
10
,
v
11
,
v
13
,
v
14
}
3
13
16
8
6
14
{
v
2
,
v
3
,
v
5
,
v
6
,
v
8
,
v
9
,
v
11
,
v
12
,
v
14
,
v
15
}
3
13
17
9
6
15
{
v
2
,
v
3
,
v
5
,
v
6
,
v
8
,
v
9
,
v
11
,
v
12
,
v
14
,
v
15
,
v
16
}
3
14
18
10
6
16
{
v
2
,
v
3
,
v
5
,
v
6
,
v
8
,
v
9
,
v
11
,
v
12
,
v
14
,
v
15
,
v
16
,
v
17
}
3
15
19
10
6
16
{
v
2
,
v
3
,
v
5
,
v
6
,
v
8
,
v
9
,
v
11
,
v
12
,
v
14
,
v
15
,
v
17
,
v
18
}
3
15Theorem 9.
(20)
D
I
(
S
′
(
P
n
)
)
=
γ
(
S
′
(
P
n
)
)
+
6
,
∀
n
≥
20
.Proof.
Letv1,v2,…,vn be the vertices of path Pn and u1,u2,…,un be the vertices corresponding to v1,v2,…,vn which are added to obtain S′(Pn). Then from Theorem 7(21)γ(S′(Pn))={n2ifn≡0(mod4)n+12ifn≡1,3(mod4)n+22ifn≡2(mod4),D={v4j+2,v4j+3} where 0≤j<⌊n/4⌋ for n≡0(mod4), D={v4j+2,v4j+3,vn-1} where 0≤j<⌊n/4⌋-1 for n≡1(mod4), D={v4j+2,v4j+3,vn-2,vn-1} where 0≤j<⌊n/4⌋ for n≡2,3(mod4) are γ-sets of S′(Pn). Then m(S′(Pn)-D)=6.
Therefore(22)DI(S′(Pn))≤γ(S′(Pn))+6.
Now we claim that if m(G-S)≠6 for any dominating set S other than D then
(23)|S|+m(S′(Pn)-S)≥γ(S′(Pn))+6.
Case
1. If S1 is any dominating set other than D and m(S′(Pn)-S1)>4 or 5 then |S1|≥γ(S′(Pn))+2. Consequently |S1|+m(S′(Pn)-S1)≥γ(S′(Pn))+6.
Case
2. If S2 is any dominating set other than D and S1 of S′(Pn) with m(S′(Pn)-S2)=3 then |S2|≥(n-⌈n/3⌉). Consequently |S2|+m(S′(Pn)-S2)≥(n-⌈n/3⌉)+3>(⌈n/2⌉+1)+6=γ(S′(Pn))+6.
Case
3. If S3 is any dominating set other than D, S1 and S2 of S′(Pn) with m(S′(Pn)-S3)=2 then |S3|≥n-2. Consequently |S3|+m(S′(Pn)-S3)≥(n-2)+2=n>(⌈n/2⌉+1)+6=γ(S′(Pn))+6.
Case
4. If S4 is any dominating set other than D, S1, S2 and S3 of S′(Pn) with m(S′(Pn)-S4)=1 then |S4|≥n. Consequently |S4|+m(S′(Pn)-S4)≥n+1>(⌈n/2⌉+1)+6=γ(S′(Pn))+6.
Thus from (22) and (23) we have DI(S′(Pn))=γ(S′(Pn))+6.Theorem 10.
For alln≥3(24)γ(S′(Cn))={n2ifn≡0(mod4)n+12ifn≡1,3(mod4)n+22ifn≡2(mod4).Proof.
Letv1,v2,…,vn be the vertices of cycle Cn and u1,u2,…,un be the vertices corresponding to v1,v2,…,vn which are added to obtain S′(Cn). Now either of the vertex from vi-1 and vi+1 must belong to any dominating set S as N(ui)={vi-1,vi+1}.
Consequently(25)|S|≥n2foranydominatingsetS.
Now depending upon the different possibilities of n we choose S as follows.Forn≡0(mod4), S={v1,v4j,v4j+1,vn} with |S|=n/2 (where 1≤j<⌊n/4⌋),forn≡1(mod4), S={v1,v4j,v4j+1} with |S|=(n+1)/2 (where 1≤j≤⌊n/4⌋),forn≡2(mod4), S={v1,v4j,v4j+1,vn} with |S|=(n+2)/2 (where 1≤j≤⌊n/4⌋),forn≡3(mod4), S={v1,v4j,v4j+1,vn} with |S|=(n+1)/2 (where 1≤j≤⌊n/4⌋).We claim that each S is a minimal dominating set of S′(Cn) since N(v1)={v2,vn,u2,un}, N(v4j)={v4j-1,v4j+1,u4j-1,u4j+1}, N(v4j+1)={v4j,v4j+2,u4j,u4j+2} and N(vn)={v1,vn-1,u1,un-1} and removal of v4j, a vertex u4j+1 will not be dominated by any vertex hence from (25):
(26)γ(S′(Cn))={n2ifn≡0(mod4)n+12ifn≡1,3(mod4)n+22ifn≡2(mod4).Theorem 11.
(27)
D
I
(
S
′
(
C
n
)
)
=
{
n
+
1
;
3
≤
n
≤
6
n
-
⌊
n
3
⌋
+
3
;
7
≤
n
≤
18
.Proof.
Letv1,v2,…,vn be the vertices of cycle Cn and u1,u2,…,un be the vertices corresponding to v1,v2,…,vn which are added to obtain S′(Cn).
Case
1
(
n
=
3
). From Theorem 10  γ(S′(C3))=2 and D={v1,v2} is a γ-set of S′(C3) then m(S′(C3)-D)=3. Therefore
(28)DI(S′(C3))≤γ(S′(C3))+m(S′(C3)-D)=2+3=5.
If S1 is any dominating set other than D of S′(C3) with m(S′(C3)-S1)=2 then |S1|≥3. This implies that
(29)|S1|+m(S′(C3)-S1)≥2+3=5.
Let S2={v1,v2,v3} be dominating set of S′(C3) and m(S′(C3)-S2)=1. This implies that
(30)|S2|+m(S′(C3)-S2)=3+1=4.
Hence from (28), (29), and (30) DI(S′C3)=4.
Case
2
(
n
=
4
). From Theorem 10  γ(S′(C4))=2 and D={v1,v2} is a γ-set of S′(C4) then m(S′(C4)-D)=3.
Therefore(31)DI(S′(C4))≤γ(S′(C4))+m(S′(C4)-D)=2+3=5.
If S1 is any dominating set other than D of S′(C4) and m(S′(C4)-S1)=2 or 1 then |S1|≥4. Therefore
(32)|S1|+m(S′(C4)-S1)≥4+2=6.
Hence from (31) and (32) DI(S′(C4))=5.
Case
3
(
n
=
5
). From Theorem 10  γ(S′(C5))=3 and D={v2,v3,v4} is a γ-set of S′(C5) then m(S′(C5)-D)=6.
Therefore(33)DI(S′(C5))≤γ(S′(C5))+m(S′(C5)-D)=3+6=9.
If S1 is any dominating set other than D of S′(C5) and m(S′(C5)-S1)=4 or 5 then |S1|≥γ(S′(C5))=3.
This implies(34)|S1|+m(S′(C5)-S1)≥γ(S′(C5))+4=3+4=7.
If S2 is any dominating set other than D and S1 with m(S′(C5)-S2)=2 or 3 then |S2|≥5. This implies that
(35)|S2|+m(S′(C5)-S2)≥5+2=7.
Let S3={v1,v2,v3,v4,v5} be dominating set of S′(C5) then m(S′(C5)-S3)=1. This implies that
(36)|S3|+m(S′(C5)-S3)=5+1=6.
Hence from (33), (34), (35), and (36) DI(S′(C5))=6.
Case
4
(
n
=
6
). From Theorem 10  γ(S′(C6))=4 and D={v1,v2,v4,v5} is a γ-set of S′(C6) then m(S′(C6)-D)=3.
Therefore(37)DI(S′(C6))≤γ(S′(C6))+m(S′(C6)-D)=4+3=7.
If S1 is any dominating set other than D of S′(C6) and m(S′(C6)-S1)=2 then |S1|≥6. This implies that
(38)|S1|+m(S′(C6)-S1)≥6+2=8.
Let S2={v1,v2,v3,v4,v5,v6} be dominating set of S′(C6) then m(S′(C6)-S2)=1.
Therefore(39)|S2|+m(S′(C6)-S2)=6+1=7.
Hence from (37), (38), and (39) DI(S′(C6))=7.
Case
5
(
7
≤
n
≤
18
). We know that
(40)DI(S′(Cn))≤γ(S′(Cn))+m(S′(Cn)-D).
The domination number with γ(S′(Cn))+m(S′(Cn)-D) and set S with |S|+m(S′(Cn)-S) are shown in Table 2.
From Table2 for any dominating set S other than D of S′(Cn), we have
(41)|S|+m(S′(Cn)-S)≤γ(S′(Cn))+m(S′(Cn)-D).
If S1 is any dominating set of S′(Cn) other than D and S with m(S′(Cn)-S1)=4 or 5 then |S|≤|S1|.
This implies that(42)|S|+m(S′(Cn)-S)≤|S1|+m(S′(Cn)-S1).
If S2 is any dominating set of S′(Cn) other than D, S, and S1 with m(S′(Cn)-S2)=2 or 1 then |S2|≥n.
This implies that(43)|S2|+m(S′(Cn)-S2)≥n+1.
Hence from Table 2 and the results (40) to (43) we conclude that
(44)DI(S′(Cn))=|S|+m(S′(Cn)-S)=n-⌊n3⌋+3.
Hence the result.Table 2
n
γ
(
S
′
(
C
n
)
)
m
(
S
′
(
C
n
)
-
D
)
γ
(
S
′
(
C
n
)
)
+
m
(
S
′
(
C
n
)
-
D
)
S
m
(
S
′
(
C
n
)
-
S
)
|
S
|
+
m
(
S
′
(
C
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7
4
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{
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3
8
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{
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9
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{
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{
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10
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{
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{
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{
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{
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8
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10
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14
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3
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{
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1
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2
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5
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7
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8
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10
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11
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14
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3
13
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8
6
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{
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1
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4
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3
14Theorem 12.
For alln≥17DI(S′(Cn))=γ(S′Cn)+6.Proof.
Letv1,v2,…,vn be the vertices of cycle Cn and u1,u2,…,un be the vertices corresponding to v1,v2,…,vn which are added to obtain S′(Cn). Then from Theorem 10(45)γ(S′(Cn))={n2ifn≡0(mod4)n+12ifn≡1,3(mod4)n+22ifn≡2(mod4),D={v1,v4j,v4j+1,vn} (where 1≤j<⌊n/4⌋) for n≡0(mod4), D={v1,v4j,v4j+1} (where 1≤j≤⌊n/4⌋) for n≡1(mod4), D={v1,v4j,v4j+1,vn} (where 1≤j≤⌊n/4⌋) for n≡2(mod4) or n≡3(mod4) are γ-sets of S′(Cn). Then m(S′(Cn)-D)=6.
Therefore(46)DI(S′(Cn))≤γ(S′(Cn))+6.
Now we claim that if m(S′(Cn))≠6 for any dominating set S other than D then
(47)|S|+m(S′(Cn)-S)≥γ(S′(Cn))+6.
Case
1. If S1 is any dominating set other than D and m(S′(Cn)-S1)=4 or 5 then |S1|≥γ(S′(Cn))+2. Consequently |S1|+m(S′(Cn)-S1)≥γ(S′(Cn))+6.
Case
2. If S2 is any dominating set other than D and S1 of S′(Cn) with m(S′(Cn)-S2)=3 then |S2|≥(n-⌊n/3⌋). Consequently |S2|+m(S′(Cn)-S2)≥(n-⌊n/3⌋)+3>γ(S′(Cn))+6.
Case
3. If S3 is any dominating set other than D, S1, and S2 of S′(Cn) with m(S′(Cn)-S3)=2 or 1 then |S3|≥n. Consequently |S3|+m(S′(Cn)-S3)≥n+1>γ(S′(Cn))+6.
Thus from (46) and (47) we have DI(S′(Cn)).


## 3. Concluding Remarks

The domination and vulnerability of network are two important concepts for the network security. We have studied an important measure of vulnerability known as domination integrity and investigate domination integrity of splitting graphs of path and cycle. The results reported here throw some light in the direction to find the domination integrity of larger graph obtained from the given graph.



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*Source: 795427-2013-01-17.xml*