Ar25260265

G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.260-265
Complementary Perfect Triple Connected Domination Number of
a Graph
G. Mahadevan*, Selvam Avadayappan**, A.Mydeen bibi***,
T.Subramanian****
*
Dept. of Mathematics, Anna University of Technology, Tirunelveli-627 002.
**
Dept.of Mathematics, VHNSN College, Virudhunagar.
***
Research Scholar, Mother Teresa Women’s University, Kodaikanal.
****
Research Scholar, Anna University of Technology Tirunelveli, Tirunelveli.

Abstract: The number of components of G is denoted by 𝜔 (G).
The concept of triple connected graphs The complement 𝐺 of G is the graph with vertex set
with real life application was introduced in [9] by V in which two vertices are adjacent if and only if
considering the existence of a path containing any they are not adjacent in G. A tree is a connected
three vertices of G. A graph G is said to be triple acyclic graph. A bipartite graph (or bigraph) is a
connected if any three vertices lie on a path in G. graph whose vertices can be divided into two disjoint
In [3], the concept of triple connected dominating sets U and V such that every edge connects a vertex
set was introduced. A set S V is a triple in U to one in V. A complete bipartite graph is a
connected dominating set if S is a dominating set special kind of bipartite graph where every vertex of
of G and the induced sub graph <S> is triple the first set is connected to every vertex of the second
connected. The triple connected domination set. The complete bipartite graph with partitions of
number 𝛾 tc(G) is the minimum cardinality taken order |V1|=m and |V2|=n, is denoted Km,n. A star,
over all triple connected dominating sets in G. In denoted by K1,p-1 is a tree with one root vertex and p –
this paper, we introduce a new domination 1 pendant vertices. A bistar, denoted by B(m, n) is
parameter, called Complementary perfect triple the graph obtained by joining the root vertices of the
connected domination number of a graph. A set stars K1,m and K1,n.
S V is a complementary perfect triple connected The friendship graph, denoted by Fn can be
dominating set if S is a triple connected constructed by identifying n copies of the cycle C3 at
dominating set of G and the induced sub graph a common vertex. A wheel graph, denoted by Wp is a
<V - S> has a perfect matching. The graph with p vertices, formed by connecting a single
complementary perfect triple connected vertex to all vertices of an (p-1) cycle. A helm graph,
denoted by Hn is a graph obtained from the wheel Wn
domination number 𝛾 cptc(G) is the minimum by joining a pendant vertex to each vertex in the
cardinality taken over all complementary perfect outer cycle of Wn by means of an edge. Corona of
triple connected dominating sets in G. We two graphs G1 and G2, denoted by G1 G2 is
determine this number for some standard classes
of graphs and obtain some bounds for general the disjoint union of one copy of G1 and |V1| copies of
graph. Their relationships with other graph G2 (|V1| is the number of vertices in G1) in which ith
theoretical parameters are investigated. vertex of G1 is joined to every vertex in the ith copy
of G2. For any real number , denotes the largest
Key words: Domination Number, Triple connected integer less than or equal to . If S is a subset of V,
graph, Complementary perfect triple connected then <S> denotes the vertex induced subgraph of G
domination number. induced by S. The open neighbourhood of a set S of
AMS (2010): 05C 69 vertices of a graph G, denoted by N(S) is the set of all
vertices adjacent to some vertex in S and N(S) S is
1 Introduction
By a graph we mean a finite, simple, called the closed neighbourhood of S, denoted by
connected and undirected graph G (V, E), where V N[S]. The diameter of a connected graph is the
denotes its vertex set and E its edge set. Unless maximum distance between two vertices in G and is
otherwise stated, the graph G has p vertices and q denoted by diam (G). A cut – vertex (cut edge) of a
edges. Degree of a vertex v is denoted by d (v), the graph G is a vertex (edge) whose removal increases
maximum degree of a graph G is denoted by Δ(G). the number of components. A vertex cut, or
We denote a cycle on p vertices by Cp, a path on p separating set of a connected graph G is a set of
vertices by Pp, and a complete graph on p vertices by vertices whose removal renders G disconnected. The
Kp. A graph G is connected if any two vertices of G connectivity or vertex connectivity of a graph G,
are connected by a path. A maximal connected denoted by κ(G) (where G is not complete) is the size
subgraph of a graph G is called a component of G. of a smallest vertex cut. A connected subgraph H of a

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connected graph G is called a H -cut if (G – H) ≥ Theorem 1.1 [9] A tree T is triple connected if and
2. The chromatic number of a graph G, denoted by only if T Pp; p ≥ 3.
χ(G) is the smallest number of colors needed to Theorem 1.2 [9] A connected graph G is not triple
colour all the vertices of a graph G in which adjacent connected if and only if there exists a H -cut with
vertices receive different colour. Terms not defined (G – H) ≥ 3 such that = 1 for at
here are used in the sense of [2].
least three components C1, C2, and C3 of G – H.
A subset S of V is called a dominating set of
Notation 1.3 Let G be a connected graph with m
G if every vertex in V − S is adjacent to at least one
vertices v1, v2, …., vm. The graph G(n1 , n2 ,
vertex in S. The domination number (G) of G is the
minimum cardinality taken over all dominating sets n3 , …., nm ) where ni, li ≥ 0 and 1 ≤ i ≤ m, is
in G. A dominating set S of a connected graph G is
obtained from G by attaching n1 times a pendant
said to be a connected dominating set of G if the
induced sub graph <S> is connected. The minimum vertex of on the vertex v1, n2 times a pendant
cardinality taken over all connected dominating set is vertex of on the vertex v2 and so on.
the connected domination number and is denoted by
Example 1.4 Let v1, v2, v3, v4, be the vertices of K4,
c. A subset S of V of a nontrivial graph G is said to
the graph K4 (2P2, P3, P4, P3) is obtained from K4 by
be Complementary perfect dominating set, if S is a
dominating set and the subgraph induced by <V-S> attaching 2 times a pendant vertex of P 2 on v1, 1 time
has a perfect matching. The minimum cardinality a pendant vertex of P3 on v2, 1 times a pendant vertex
of P4 on v3 and 1 time a pendant vertex of P 3 on v4
taken over all Complementary perfect dominating
and is shown in Figure 1.1.
sets is called the Complementary perfect domination
number and is denoted by cp.
One can get a comprehensive survey of
results on various types of domination number of a v1 v2
graph in [11].
Many authors have introduced different
types of domination parameters by imposing v4 v3
conditions on the dominating set [1,8]. Recently the
concept of triple connected graphs was introduced by
Paulraj Joseph J. et. al.,[9] by considering the
Figure 1.1: K4 (2P2, P3, P4, P3)
existence of a path containing any three vertices of G.
They have studied the properties of triple connected
graph and established many results on them. A graph 2 Complementary Perfect Triple Connected
G is said to be triple connected if any three vertices Domination Number
lie on a path in G. All paths and cycles, complete Definition 2.1 A set S V is a complementary
graphs and wheels are some standard examples of perfect triple connected dominating set if S is a triple
triple connected graphs. connected dominating set of G and the induced sub
A set S V is a triple connected dominating graph <V - S> has a perfect matching. The
set if S is a dominating set of G and the induced sub complementary perfect triple connected domination
graph <S> is triple connected. The triple connected number cptc(G) is the minimum cardinality taken
domination number tc(G) is the minimum over all complementary perfect triple connected
cardinality taken over all triple connected dominating dominating sets in G. Any triple connected
sets in G. dominating set with cptc vertices is called a cptc -set
A set S V is a complementary perfect of G.
triple connected dominating set if S is a triple Example 2.2 For the graph G1 in Figure 2.1, S = {v1,
connected dominating set of G and the induced sub v2, v5} forms a cptc -set of G1. Hence cptc (G1) = 3.
graph <V - S> has a perfect matching. The
complementary perfect triple connected domination
v1
number cptc(G) is the minimum cardinality taken
over all complementary perfect triple connected
dominating sets in G.
G 1: v5 v2
In this paper we use this idea to develop the
concept of complementary perfect triple connected v3
v4
dominating set and complementary perfect triple
connected domination number of a graph. Figure 2.1
Observation 2.3 Complementary perfect triple
connected dominating set does not exists for all
Previous Results graphs and if exists, then cptc ≥ 3.

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Example 2.4 For the graph G2 in Figure 2.2, any
v6 v1
minimum dominating set must contain the supports
and any connected dominating set containing these
supports is not Complementary perfect triple
connected and hence cptc does not exists. K6: v5 v2

G 2:
v4 v3

Figure 2.2 Figure 2.5

Remark 2.5 Throughout this paper we consider only Theorem 2.12 If the induced subgraph of each
connected graphs for which complementary perfect connected dominating set of G has more than two
triple connected dominating set exists. pendant vertices, then G does not contain a
Observation 2.6 The complement of the Complementary perfect triple connected dominating
complementary perfect triple connected dominating set.
set need not be a complementary perfect triple Proof This theorem follows from Theorem 1.2.
connected dominating set. Example 2.13 For the graph G6 in Figure 2.6,
Example 2.7 For the graph G3 in Figure 2.3, S = {v6, v2, v3, v4} is a minimum connected
= {v1, v2, v3} forms a Complementary perfect triple dominating set so that c (G6) = 4. Here we notice
connected dominating set of G3. But the complement that the induced subgraph of S has three pendant
V – S = {v4, v5, v6, v7} is not a Complementary perfect vertices and hence G does not have a Complementary
triple connected dominating set. perfect triple connected dominating set.
v7
v1 v3
v2 v6
G 6:
v7
v4 v1 v2
v6 v3 v4 v5
G 3: v5
Figure 2.6
Figure 2.3
Observation 2.8 Every Complementary perfect triple Complementary perfect triple connected
connected dominating set is a dominating set but not domination number for some standard graphs are
the converse. given below.
Example 2.9 For the graph G4 in Figure 2.4, S = {v1} 1) For any cycle of order p ≥ 5, cptc (Cp) = p – 2.
is a dominating set, but not a complementary perfect 2) For any complete bipartite graph of order p ≥ 5,
triple connected dominating set.
cptc (Km,n) =
v1 (where m, n ≥ 2 and m + n = p.
3) For any complete graph of order p ≥ 5,
G 4: v4
v2 cptc (Kp) =
v3 4) For any wheel of order p ≥ 5,
Figure 2.4
Observation 2.10 For any connected graph G, (G) cptc (Wp) =
≤ cp(G) ≤ cptc(G) and the inequalities are strict. 5) For any Fan graph of order p ≥ 5,
Example 2.11 For K6 in Figure 2.5, (K6) = {v1} = 1,
cp(K6) = {v1, v2}= 2 and cptc(K6) = {v1, v2, v3, v4}= cptc (Fp) =
4. Hence (G) ≤ cp(G) ≤ cptc(G). 6) For any Book graphs of order p ≥ 6,
cptc (Bp) = 4.
7) For any Friendship graphs of order p ≥ 5,
cptc (Fn)) = 3.
Observation 2.14 If a spanning sub graph H of a
graph G has a Complementary perfect triple

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connected dominating set, then G also has a Proof The lower and upper bounds trivially follows
complementary perfect triple connected dominating from Definition 2.1. For C5, the lower bound is
set. attained and for C9 the upper bound is attained.
Example 2.16 For any graph G and H in Figure 2.7, Theorem 2.20 For a connected graph G with 5
S = {v1, v4, v5} is a complementary perfect triple vertices, cptc(G) = p – 2 if and only if G is
connected dominating set and so cptc (G) = 3. For the isomorphic to C5, W5, K5, K2,3, F2, K5 – {e}, K4(P2),
spanning subgraph H, S = {v1, v4, v5} is a C4(P2), C3(P3), C3(2P2) or any one of the graphs
complementary perfect triple connected dominating shown in Figure 2.9.
set and so cptc(H) = 3. H1: x H2: x
V1
G:
v y v y
V5
V4 V2 z u z u

H3: y H4: y

V3
H: V1 x z x u

u v z v

V4
V5 V2 H5: x H6: x

v y v y
V3 u z u z
Figure 2.7
H7: x
Observation 2.17 Let G be a connected graph and H
be a spanning sub graph of G. If H has a
complementary perfect triple connected dominating
v y
set, then cptc (G) ≤ cptc (H) and the bound is sharp.
Example 2.18 For the graph G in Figure 2.8, u z
S = {v1, v2, v7, v8} is a complementary perfect triple
connected dominating set and so cptc (G) = 4. For the Figure 2.9
spanning subgraph H of G, S = {v1, v2, v3, v6, v7, v8} Proof Suppose G is isomorphic to C5, W5, K5, K2,3,
is a complementary perfect triple connected F2, K5 – {e}, K4(P2), C4(P2), C3(P3), C3(2P2) or any
dominating set and so cptc (H) = 6. one of the graphs H1 to H7 given in Figure 2.9, then
clearly cptc (G) = p – 2. Conversely, let G be a
G: V1 V2 V3 V4
connected graph with 5 vertices and cptc (G) = 3.
Let S = {x, y, z} be a cptc -set, then clearly <S> = P3
or C3. Let V – S = V (G) – V(S) = {u, v}, then
<V – S> = K2 = uv.
Case (i) <S> = P3 = xyz.
Since G is connected, there exists a vertex say x (or y,
V8 V7 V6 V5 z) in P3 is adjacent to u (or v) in K2, then cptc -set of G
does not exists. But on increasing the degrees of the
7
V1 V2 V3 V4 vertices of S, let x be adjacent to u and z be adjacent
H:
to v. If d(x) = d(y) = d(z) = 2, then G  C5. Now by
increasing the degrees of the vertices, by the above
argument, we have G  K5, K5 – {e}, K4(P2), C4(P2),
C3(P3) or any one of the graphs H1 to H7 given in
Figure 2.9. Since G is connected, there exists a vertex
say y in P3 is adjacent to u (or v) in K2, then cptc -set
V8 V7 V6 V5
of G does not exists. But on increasing the degrees of
Figure 2.8 the vertices of S, let y be adjacent to v, x be adjacent
Theorem 2.19 For any connected graph G with p ≥ 5, to u and v and z be adjacent to u and v. If d(x) = 3,
we have 3  cptc(G) ≤ p - 2 and the bounds are sharp.

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d(y) = 3, d(z) = 3, then G  W5. Now by increasing 3. Complementary Perfect Triple Connected
the degrees of the vertices, by the above argument, Domination Number and Other Graph
we have G  K2,3, C3(2P2). In all the other cases, no Theoretical Parameters
new graph exists. Theorem 3.1 For any connected graph G with p ≥ 5
Case (ii) <S> = C3 = xyzx. vertices, cptc (G) + κ(G) ≤ 2p – 3 and the bound is
Since G is connected, there exists a vertex say x (or y, sharp if and only if G  K5.
z) in C3 is adjacent to u (or v) in K2, then S = {x, u, v} Proof Let G be a connected graph with p ≥ 5 vertices.
forms a cptc -set of G so that cptc (G) = p – 2. If We know that κ(G) ≤ p – 1 and by Theorem 2.19, cptc
d(x) = 3, d(y) = d(z) = 2, then G  C3(P3). If d(x) = 4, (G) ≤ p – 2. Hence cptc (G) + κ(G) ≤ 2p – 3. Suppose
d(y) = d(z) = 2, then G  F2. In all the other cases, no G is isomorphic to K5. Then clearly cptc (G) + κ (G)
new graph exists. = 2p – 3. Conversely, Let cptc (G) + κ(G) = 2p – 3.
Nordhaus – Gaddum Type result: This is possible only if cptc (G) = p – 2 and κ (G) =
Theorem 2.21 Let G be a graph such that G and p – 1. But κ (G) = p – 1, and so G  Kp for which cptc
have no isolates of order p ≥ 5, then (G) = 3 = p – 2 so that p = 5. Hence G  K5.
(i)cptc (G) + cptc ( ) ≤ 2(p – 2) Theorem 3.2 For any connected graph G with p ≥ 5
(ii) cptc (G) . cptc ( )≤ (p – 2)2 and the bounds are vertices, cptc (G) +  (G) ≤ 2p – 2 and the bound is
sharp if and only if G  K5.
sharp. Proof Let G be a connected graph with p ≥ 5 vertices.
Proof The bounds directly follow from Theorem
We know that (G) ≤ p and by Theorem 2.19,
2.19. For the cycle C5, cptc (G) + cptc ( ) = 2(p – 2) cptc(G) ≤ p – 2. Hence cptc (G) + (G) ≤ 2p – 2.
and cptc (G). cptc ( )≤ (p – 2)2. Suppose G is isomorphic to K5. Then clearly
Theorem 2.22 γcptc (G) ≥  p / ∆+1  cptc (G) + (G) = 2p – 2. Conversely, Let cptc (G)
Proof Every vertex in V-S contributes one to degree + (G) = 2p – 2. This is possible only if cptc (G) =
sum of vertices of S. Then |V - S|   u  S d(u) where p – 2 and (G) = p. But (G) = p, and so G is
S is a complementary perfect triple connected isomorphic to Kp for which cptc (G) = 3 = p – 2 so
dominating set. So |V - S|  γcptc ∆ which implies that p = 5. Hence G  K5.
(|V|- |S|)  γcptc ∆. Therefore p - γcptc  γcptc ∆, which Theorem 3.3 For any connected graph G with p ≥ 5
implies γcptc (∆+1) ≥ p. Hence γcptc (G) ≥  p / ∆+1  vertices, cptc (G) +(G) ≤ 2p – 3 and the bound is
Theorem 2.23 Any complementary perfect triple sharp if and only if G is isomorphic to W5, K5, C3(2),
connected dominating set of G must contains all the K5 – {e}, K4(P2), C3(2P2) or any one of the graphs
pendant vertices of G. shown in Figure 3.1.
Proof Let S be any complementary perfect triple
connected dominating set of G. Let v be a pendant G 1: v4 G 2: v4
vertex with support say u. If v does not belong to S,
then u must be in S, which is a contradiction S is a
complementary perfect triple connected dominating v3 v v3 v
set of G. Since v is a pendant vertex, so v belongs to
S. v2 v1 v2 v1
Observation 2.24 There exists a graph for which G 3: v4 v4
γcp (G) = γtc(G) = γcptc (G) is given below.
G 4:

v1 v2 v3 v v3 v
v6
v2 v1 v2 v1
v5 Figure 3.1
Proof Let G be a connected graph with p ≥ 5 vertices.
v4
We know that (G) ≤ p – 1 and by Theorem 2.19,
cptc (G) ≤ p – 2. Hence cptc (G) + (G) ≤ 2p – 3. Let
v3 v7 G be isomorphic to W5, K5,C3(2), K5 – {e}, K4(P2),
C3(2P2) or any one of the graphs G1 to G4 given in
H: Figure 3.1, then clearly cptc (G) +(G) = 2p – 3.
Figure 2.10 Conversely, Let cptc (G) + (G) = 2p – 3. This is
For the graph Hi in figure 2.10, S = {v1, v3, v7} is a possible only if cptc (G) = p – 2 and (G) = p – 1.
triple connected dominating set complementary But cptc (G) = p – 2 and (G) = p – 1, by
perfect and complementary perfect triple connected Theorem 2.20, we have G  W5, K5, C3(2), K5 – {e},
dominatin set. Hence γcp (G) = γtc(G) = γcptc (G) = 3. K4(P2), C3(2P2) and the graphs G1 to G4 given in
Figure 3.1.

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REFERENCES
[1] Cokayne E. J. and Hedetniemi S. T. (1980):
Total domination in graphs, Networks,
Vol.10: 211–219.
[2] John Adrian Bondy, Murty U.S.R. (2009):
Graph Theory
[3] Mahadevan G. SelvamAvadayappan,
Paulraj Joseph. J, Subramanian T: (2012)
Tripleconnected domination number of a
graph – Preprint
[4] Nordhaus E. A. and Gaddum J. W. (1956):
On complementary graphs, Amer. Math.
Monthly, 63: 175–177.
[5] Paulraj Joseph J. and Arumugam. S.
(1992): Domination and connectivity in
graphs, International Journal of
Management Systems, 8 (3): 233–236.
(1995): Domination in graphs,
International Journal of Management
Systems, 11: 177–182.
(1997): Domination and coloring in
graphs, International Journal of
Management Systems, 8 (1): 37–44.
[8] Paulraj Joseph J. , Mahadevan G. and
Selvam A. (2006): On Complementary
perfect domination number of a graph, Acta
Ciencia Indica, Vol. XXXI M, No. 2, 847 –
854.
[9] Paulraj Joseph J., Angel Jebitha M.K.,
Chithra Devi P. and Sudhana G. (2012):
Triple connected graphs, (To appear)
Indian Journal of Mathematics and
Mathematical Sciences.
[10] Sampathkumar, E.; Walikar, HB (1979): The
connected domination number of a graph,
J. Math. Phys. Sci 13 (6): 607–613.
[11] Teresa W. Haynes, Stephen T. Hedetniemi
and Peter J. Slater (1998): Domination in
graphs, Advanced Topics, Marcel Dekker,
New York.
[12] Teresa W. Haynes, Stephen T. Hedetniemi
and Peter J. Slater (1998): Fundamentals of
domination in graphs, Marcel Dekker, New
York.

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