J. Comp. & Math. Sci. Vol. 1(5), 598-605 (2010)
Domination in Semitotal-Point Graph B. BASAVANAGOUD*, S.M.HOSAMANI and S.H.MALGHAN Department of Mathematics, Karnatak University,Dharwad-580 003, Karnataka, India e-mail*: b.basavanagoud@gmail.com bgouder1@yahoo.co.in *Corresponding author. ABSTRACTS A set D of vertices in a graph G is a dominating set of G if every vertex not in D is adjacent to atleast one vertex in D. The domination number
(G) of G is the cardinality of a minimum dominating set of G. The semitotal-point graph T2 (G) of G is the graph whose vertex set is
V (G ) E (G) , where two vertices are adjacent if and only if (i) they are adjacent vertices of G or (ii) one is a vertex of G and the other is an edge of G, incident with it. In this paper, some results on domination number of semitotal-point graph are obtained. 2000 Mathematics Subject Classification: 05C69. Keywords: semitotal-point graph, domination number, connected domination number, total domination number.
INTRODUCTION The graphs considered here are finite, undirected without loops or multiple edges. Any undefined term in this paper may be found in Harary3. Let G = (V , E ) be a graph with V = p and E = q . In a graph G if deg(v) = 1, then v is called an end vertex or a pendant vertex of G. Each
vertex v V dominates every vertex in its s closed neighbourhood N [v] . A set S V is a dominating set if each vertex in V is dominated by atleast one vertex of S. The dominating number (G) is the smallest cardinality of a dominating set. A dominating set D of a graph G is a split dominating set if the induced subgraph V D is disconnected. The
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B. Basavanagoud et al., J. Comp. & Math. Sci. Vol. 1(5), 598-605 (2010)
split domination number s (G ) of G is the minimum cardinality of a split dominating set. A dominating set D of a graph G is a connected dominating set if the induced D is connected. The subgraph connected domination number c (G ) of G is the minimum cardinality of a connected dominating set. A dominating set D is a total dominating set if the induced subgraph D has no isolated vertex. The total domination number t (G ) of a graph G is the minimum cardinality of a total dominating set. A set D of vertices of a graph G is a maximal dominating set of G if D is a dominating set and V D is not a dominating set. The maximal domination number m (G ) of G is the minimum cardinality of a maximal dominating set. A dominating set D of G is a cototal dominating set if the induced subgraph V D has no isolated vertices. The cototal domination number cot (G ) of G is the minimum cardinility of a cototal dominating set. A dominating set D of vertices in a graph G is a dominating clique if the induced subgraph D is a complete graph. The clique domination number
in G. The path nonsplit domination number
pns (G ) is the minimum cardinality of a path nonsplit dominating set. The maximum number of classes of a domatic partition of G is called the domatic number of G and is denoted by d(G). The vertex independence number
0 (G ) is the maximum cardinality among the independent set of vertices of G. E.Sampatkumar and S. B. Chikkodimath6 introduced the new graph valued function, namely the semitotal-point graph of a graph. The semitotal-point graph T2(G) of a graph G is the graph whose vertex set is V (G ) E (G) , where two vertices are adjacent if and only if (i) they are adjacent vertices of G or (ii) one is a vertex of G and the other is an edge of G, incident with it. In Figure 1, a graph and its semitotalpoint graph are shown.
4
4
a
a
1
G:
1
T2(G) : d
b
b
d
cl (G ) of G is the minimum cardinality of a dominating clique. A dominating set D of a connected graph G is a path nonsplit dominating set if the induced subgraph V D is a path
2
c
3
3
2
Figure 1
c
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PRELIMINARY RESULTS We need the following Theorems for our further results. Theorem A[4]. For any graph G with a pendant vertex (G ) = s (G ). Theorem B[1]. If G is a connected graph
2p . 3 Theorem C[7]. If T is a tree with p 3 with p 3 vertices, then t (G)
vertices, then c (T ) = p e. where e denotes the number of pendant vertices in T. Theorem D[3]. For any nontrivial connected
( p, q )
graph
p
vertices, (T2 (C p )) = . 2 Proposition 3. For any wheel W P with
p p 4 vertices, (T 2 (W p )) = 2 1. Proposition 4. For any star K 1, p 1 with
p 2 vertices, (T 2 ( K 1, p 1 )) = 1. Proposition 5. For any KP with p 2 vertices, (T 2 ( K p )) = p 1. Proposition 6. If G = K p , then G T2 (G ) . MAIN RESULTS
G
0 0 = p = 1 1 . Theorem E[8]. For any connected ( p, q )
Theorem 3.1. For any ( p , q ) graph G,
(G) (T2 (G )) . The equality holds for G = K 2 or K p .
p graph G d (G ). p (G )
Proof. We consider the following cases. Case 1. Let G be a graph without isolated
Theorem F[2]. If G is a graph without isolated vertices of order p, then
vertices.Then by Theorem F, (G )
(G)
p . 2
p . 2
Therefore by def inition of T2 (G ) ,
pq . Hence 2
Next we give the domination number of semitotal-point graphs of some standard class of graphs.
(T2 (G))
Proposition 1. For any path PP with p 2
Case 2. Let G be a graph with isolated
p vertices, (T2 ( Pp )) = . 2 Proposition 2. For any cycle CP with p 3
(G) (T2 (G )) . vertices of the type G = H nK1 , where
H = ( p, q) is any connected graph.Then T2 (G) = H nK1 . By Case 1,
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B. Basavanagoud et al., J. Comp. & Math. Sci. Vol. 1(5), 598-605 (2010)
( H ) ( H ) . Therefore (G) and (T2 (G))
p n 2
p q n . Hence the 2
601
(T2 (G)) p . The equality holds for G = K p. Proof. We consider the following cases. Case 1. Suppose G is a connected graph.
( G ) p 1 .
result. Equality can be easily verified for
Then
G = K 2 or K p .
Proposition 5, (T2 (G )) p 1 .
Theorem 3. For any nontrivial tree T,
Case 2. Let G be a disconnected graph with n components that is G = H1 H2 Hn.
(T2 (T )) (T ) . Proof. Let T be any nontrivial tree. Let
D1 = {v1 , v2 ,vr } be a subset of vertex set
Then
Therefore
by
T2 (G ) = H 1 H 2 H’ n . W e
such that degvi 2 f or 1 i r and
consider the following subcases. Subcase 2.1. Suppose each component
D2 = {u1 , u2 , u s } where degui = 1 f or
H i ; 1 i n is a complete graph of order
1 i s be the subsets of V (T ) where
V (T ) = D1 D2 . Clearly D1 is a minimum dominating
set
of
T.
2 , then by Case 1, ( H1)
Therefore
(T ) D1 = r . Let D1 = {v1, v2 ,vr } and D2 = {u1 , u2 , u s } be the corresponding
( H 2 )
p'2 2
, , ( H n )
p'n 2
p'1 , 2
where
p'1 = V ( H1) , p '2 = V ( H 2 ) , p'n = V ( H n ) .
in T2 (G )
Then (T2 (G )) = p n , where n is the
respectively. Let u1 , u2 , un E (T ) and
number of components in G. Hence
vertices of D 1 and D 2
u1 , u2 ,u n be the corresponding edge vertices in T2 (G ) . For each edge ui for
(T2 (G )) < p 1 . Subcase 2.2. Suppose the components
H i ; 1 i n , are either complete or other
1 i n there exists a vertex v j D1 .
than complete graphs then by Subcase
(T ) D1 D'1 (T2 (T )) .
2.1, (T2 (G )) < p n < p 1 .The equality
Hence the result. Corollary 1. In a tree if every nonpendant vertex is adjacent to at least one pendant
is easy to verify for G = K p . Corollary 2. For any connected
Clearly
vertex, then (T2 (T )) = (T )
( p , q ) graph G, (T2 (G)) p 1 . Proposition 7. For any graph G,
Theorem 3.3 For any ( p , q ) graph G, Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
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(T2 (G )) 0 0 . Proof. Follows from Theorem D and Theorem 3.3.
p p p = . 2 2
By
Proposition
1,
Theorem 3.4 For any ( p , q ) graph G,
p (T2 ( Pp )) = . Hence the equality.. 2
p (T2 (G )) 0 0 . 1 (G )
Case 2. If G = C p p 3 vertices then
Further the equality of lower bound is attained if G = K p . Proof. We first consider the lower bound. Let D be a dominating set of G. Each vertex dominates at most itself and other vertices
of
(G) .
Hence
p (T2 (G )) . 1 (G )
p 0 ( C p ) = . 2 p
p
Therefore p 0 (G ) = p = . 2 2
p
By Proposition 2, (T2 (C p )) = 2
If G = K p , then by Theorem 3.3,
The upper bound follows from Theorem 3.3. Corollary 3. For any graph G,
(T2 ( K p )) = p . Hence the lower bound is
0 (G ) (T2 (G )).
attained. The upper bound follows from the Proposition 7. Theorem 3.5 For any ( p , q ) graph G,
p 0 (G ) (T2 (G ) p . Further the equality of lower bound is attained if G = Pp or C p . Proof. We first consider the lower bound. It is obvious that for any graph G,
p 0 (G ) (T2 (G )) . For equality,we consider the following cases: Case 1. If G = Pp for p 2 vertices then
p 0 ( Pp ) = . Therefore p 0 (G ) = 2
Theorem 3.6 For any graph G,
(T2 (G)) diam(G) . Proof. Let A = {e1, e2,et } be the set of edges which constitutes the longest path between any two vertices of G such that
A = diam(G ) . We consider the following cases: Case 1. If G contains only one edge then
T2 (G ) = C3 . Therefore (T2 (G )) = 1 = A . Case 2. Let
u1 , u 2 , , um E (G ) and
u1 , u 2 , , um be the corresponding edge vertices of T2 (G ) . Let v1 , v2 , , vn V (G) and v1 , v2 , , vn be the corresponding
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
B. Basavanagoud et al., J. Comp. & Math. Sci. Vol. 1(5), 598-605 (2010)
vertices of T2 (G ) . Since all the edge vertices are incident with V (T2 (G)) um , therefore
(T2 (G )) V (T2 (G)) um = D A . Hence (T2 (G )) diam(G ) . Theorem 3.7 For any tree T,
(T2 (T )) s (T ) . Proof. By Theorem 3.2 and Theorem A we get the required result. Theorem 3.8 For any connected graph G of order atleast three, (T2 (G)) t (G ) . Proof. By Corollary 2 and Theorem we get the required result. Theorem 3.9 For any connected graph G,
603
maximal independent set is a minimal dominating set, therefore
D = u m = V (T2 (G )) D dominating
set
of
is
also
T2 (G ) ,
a
Thus
(T2 (G )) < m (G ) , a contradiction. Theorem 3.10 For any tree T of order
p 2 vertices, c (T ) (T2 (T )) . Proof. Let T be a tree of order p 2 . Let
D = {v1 , v2 ,, vr }
be
such
that
deg (vi ) 2 for 1 i r . Then in T2 (G ) degree of each ui and each vr will become twice. Therefore, each vi in T2 (G ) will be a minimal dominating set. Therefore
(T2 (G )) m (G ) where m (G ) is the
(T2 (T )) = p e where e is the number
maximal domination number of G. Proof. Let G be any connected graph.
of pendant vertices in T. Hence by Theorem
Suppose (T2 (G )) > m (G ) . Then each
ui , 1 i m of a corresponding edge vertex of G in T2 (G ) will be adjacent to at least two vertices of V (T2 (G ) vn ) in
T2 (G ) , where vn is the corresponding vertex set of G in T2 (G ) . Therefore by
C, c (T ) (T2 (T ) . Theorem 3.11 For any ( p , q ) graph G,
cot (T2 (G )) = 1 if and only if G = K1, p 1 . Proof. If G = K1, p 1 , then degvi = 2degvi where vi is the corresponding vertex of
vi of G in T2 (G ) . Each component of
T2 (G ) will be a triangle and each triangle
taking alternate vertices v j , 1 j n of
will have one common vertex which is
T2 (G ) , the set v j , j = 1,2,...., n will be a
adjacent to all other vertices of T2 (G ) .
dominating set of T2 (G ) . Since each ui ,
Therefore cot (T2 (G )) = 1 .
1 i m is an independent vertex in
Conversely, suppose cot (T2 (G )) = 1 and
T2 (G ) , then the set ui , i = 1,2,...., n is a
G K1, p 1 , then there exists a vertex of
maxiaml independent set. Since every Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
B. Basavanagoud et al., J. Comp. & Math. Sci. Vol. 1(5), 598-605 (2010)
604
(T2 (G)) = p 1 . Let u be a vertex of
Hence G = K p .
degree p 1 , then all other vertices of
Proposition 7. For any path Pp with p 2
T2 (G ) are adjacent to u. But in T2 (G ) ,
vertices, pns (T2 (G )) = q where q is the
this is the only case when G is a star, otherwise
(T2 (G)) < p 1 .
Thus
(T2 (G)) 2 , a contradiction. Theorem 3.12 For any ( p , q ) graph G,
cl (T2 )(G ) = 1 if and only if G = K1, p 1 . Proof. Proof is similar to the Theorem 3.11. Theorem 3.13 For any graph G,
number of edges in Pp . Proof. Let G be a path with p 2 vertices. Let u m ; 1 m q be the corresponding edge vertices in T2 (G ) and vn 1 n p is the corresponding vertices of G in
T2 (G ) , then
pns (T2 (G ) = V (T2 (G ) u m )
(T2 (G)) cot (G) . Proof. Since t (G ) cot (G ) , therefore by
= um = q .
Theorem 3.8, we get the required result.
Theorem 3.15 For any graph G,
Theorem 3.14 For any
d (G ) (T2 (G )) where d(G) is the
( p , q ) graph G,
m (T2 (G )) p .
domatic number of G.
Further equality holds if and only if
Proof. For any graph G, d (G) (G) 1 .
G = K p. Proof . For any graph G, it is easy to
Let u be a vertex of minimum degree in G, then degree of the corresponding vertex
observe that m (T2 (G )) p . Suppose
G = K p , then m (G ) = p = m (T2 (G )) . Conversely, let
m (T2 (G )) = p and
G K p . If G contains an isolated vertex then m (T2 (G )) < p . So consider a graph G without isolated vertices. Then there exists at least one vertex with (G ) 2 which will be present in both dominating set and maximal dominating set of T2 (G ) . Therefore (T2 (G )) < p , a contradiction.
ui in T2 (G ) will be twice. Also by Theorem 3.3, (T2 (G )) p . Therefore
d (G ) (T2 (G )) . If G = K p , then d ( K p ) = p = (T2 (G )) .Hence the equality holds. Theorem 3.16
p (T2 (G )) . p (G )
For any graph G,
Proof. Follows from Theorem E and Theorem 3.15. In the next Theorem we prove the
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
B. Basavanagoud et al., J. Comp. & Math. Sci. Vol. 1(5), 598-605 (2010)
Nordhaus-Gaddum type results. Theorem 3.17 For any connected ( p, q ) graph G (i) (T2 (G )) (T2 (G )) 2 p 1
3. 4.
(ii) (T2 (G )). (T2 (G )) p ( p 1) . Proof. By using Proposition 5 and Proposition 6 we get the required result. ACKNOWLEDGEMENT This research was supported by the University Grants Commission, New Delhi,India-No.F.4-3/2006(BSR)/7-101/ 2007(BSR) dated: September 2009. This research was supported by the University Grants Commission, New Delhi,India-No.F.FIP/ Plan/KAKA047TF02 dated:October 8, 2009.
5.
6.
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