Cmjv02i03p0547 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

On the Common Neighbourhood Domination Number ANWAR ALWARDI, N. D. SONER and KARAM EBADI Department of Studies in Mathematics University of Mysore, Mysore 570 006, India National Center for Advanced Research in Discrete Mathematics Kalasalingam University Tamil Nadu, India ABSTRACT We introduce the concept of common neighbourhood domination number (CN-domination number) γ cn (G ) of a graph G and analogous to this concept we define many other concepts like the CN-independence number β cn (G ) , CN-neighbourhood number

η cn (G ) , number

total CN-domination number

α cn (G )

γ tcn (G ) ,

CN-covering

and we study some relations between them,

and we also introduce some CN-graphs associate with the graph G . Keywords: Common neighbourhood domination, Common neighbourhood dominating set, Common neighbourhood graph. Mathematics Subject Classification:05C69.

1. INTRODUCTION All the graph considered here are finite and undirected with no loops and multiple edges. As usual p =| V | and q =| E | denote the number of vertices and edges of a graph G, respectively. In general, to denote the subgraph we use 〈X 〉 induced by the set of vertices X and N (v) and N [v] denote the open and closed neighbourhoods of a vertex v, respectively. A set D of vertices in a graph G is a dominating set if every vertex in V − D is adjacent to some vertex in D . The

domination number γ (G ) is the minimum cardinality of a dominating set of G . A set S ⊆ V is a neighbourhood set of G , if 〈 N [v]〉 is the G = Uv∈S 〈 N [v ]〉 , where subgraph of G induced by v and all vertices adjacent to v . The neighbourhood number η (G ) of G is the minimum cardinality of a neighbourhood set of a graph S ⊆V G . A neighbourhood set is a minimal neighbourhood set, if S − v for all v ∈ S , is not a neighbourhood set of G . A strongly regular graph with parameters (n, k , λ , µ ) is k - regular graph

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

548

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

with n vertices such that for any two λ common adjacent vertices have neighbours, and any two non- adjacent vertices have µ common neighbours.and when λ = 0 the strongly regular graph called strongly regular graph with no triangles (SRNT graph), A strongly regular graph that is connected, and its complement is connected is called primitive. For terminology and notations not specifically defined here we refer reader to3. For more details about domination number and neighbourhood number and their related and4-8. parameters, we refer to1,2 In this paper we introduce the concept of common neighbourhood domination number (CN-domination number). In ordinary domination more adjacency between vertices is enough for a vertex to dominate another in practice; neighbourliness alone between persons is not enough for friendship. If the persons have common friend then it may result in friendship. Human beings have a tendency to move with others when they have common friends. 2. CN-DOMINATING SETS Definition 2.1. Let G be simple graph with vertex set V (G ) = {v1 , v2 ,..., vn } For i ≠ j , the common neighborhood of the the vertices vi and v j , denoted by Γ(vi , v j ) , is the set of vertices, different from vi

and

v j , which are adjacent to both v i and v j . Definition 2.2. Let G = (V , E ) be a graph. A subset D of V is called common neighbourhood dominating set (CNdominating set) if every v ∈ V − D there

exist a vertex u ∈ D such that uv ∈ E (G ) and | Γ (u , v) |≥ 1 , where | Γ(u , v) | is the number of common neighbourhood between the vertices u and v . The minimum cardinality of such dominating set denoted by γ cn and is called common neighbourhood domination number (CN-domination number) of G . It is clear that CN-domination number is defined for any graph. Observation 2.3. From the definition of common neighbourhood domination number we can obtain (i)

γ cn ( K p ) = 1 , where p ≠ 2 ,

(ii)

γ cn ( Pp ) = p

(iii)

γ cn (C p ) = p

(iv)

γ cn ( K r , m ) = r + m

(v)

for any wheel graph with p vertices γ cn (W p ) = 1

Definition 2.4. A common neighbourhood dominating set D is said to be minimal common neighbourhood dominating set if no proper subset of D is common neighbourhood dominating set. Definition 2.5. A minimal common neighbourhood dominating set D of maximum cardinality is called Γcn -set and its cardinality is denoted by Γcn . Observation 2.6. Let G = (V , E ) be a graph and u ∈ V be such that | Γ(u, v) |= 0 for all v ∈ N (u ) , then u is in every common neighbourhood dominating set, such points are called common neighbourhood isolates. Let I cn denote the set of all common

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

neighbourhood isolated points of G . Hence I s ⊆ I cn ⊆ D , where I s is the set of isolated points and D is the minimum CNdominating set of G . Theorem 2.7 A common neighbourhood dominating set D is minimal if and only if for every vertex u ∈ D one of the following holds: (i) either N (u ) ∩ D = φ or | Γ(u, v) |= 0 for all v ∈ N (u ) ∩ D . (ii) there exist a vertex v ∈ V − D such that N (v) ∩ D = {u} and | Γ (u , v) |≥ 1 . Proof. Suppose that D is minimal common neighbourhood dominating set and let the conditions (i) and (ii) not hold. Then for some u ∈ D there exist v ∈ N (u ) ∩ D such that | Γ (u , v) |≥ 1 and for every v ∈V − D either N (u ) ∩ D ≠ φ or | Γ(u, v) |= 0 . Therefore D − u is common neighbourhood dominating set. and this is contradiction to the definition of minimality D . Hence (i) or (ii) holds. of Conversely, suppose for every u ∈ D one of the statement (i) or (ii) holds. Suppose D is not minimal. Then there exist u ∈ D such that D−u is common neighbor-hood dominating set. Hence there exist v ∈ D − {u} such that v is common neighbourhood dominates u . That is v ∈ N (u ) and | Γ (u , v) |≥ 1 . Therefore does not satisfy (i). The u must satisfy (ii), so there exist v ∈V − D such that and | Γ (u , v) |≥ 1 . Since N (u ) ∩ D = u D − u is common neighbourhood dominating set, there exist w ∈ D − u such that w is adjacent to v and w has common

549

neighbourhood with v. Therefore w ∈ N (v) ∩ D , | Γ(w, v) |≥ 1 and w ≠ u , a contradiction to N (v ) ∩ D = u . Therefore D is minimal common neighbourhood dominating set. Theorem 2.8. A graph G has a unique minimal common neighbourhood dominating set if and only if the set of all common neighbourhood isolates forms a common neighbourhood dominating set. Proof. Let G has a unique minimal common neighbourhood dominating set D , and suppose S = {u ∈ V : u is common neighbourhood isolate } . Then S ⊆ D , now suppose D − S ≠ φ , let v ∈ D − S , since v is not common neighbourhood isolate, V − {v} is common neighbourhood dominating set. Hence there exist a minimal common neighbourhood dominating set D1 ⊆ V − {v} and D1 ≠ D a contradiction to the fact that G has a unique minimal common neighbourhood dominating set. Conversely if the set of all common neighbourhoods isolates forms a common neighbourhood dominating set, then it is clear that G has a unique minimal common neighbourhood dominating set. Theorem 2.9. Let G be a graph without common neighbourhood isolated vertices. (That is, given any u ∈ V , there exist v ∈ V such that uv ∈ E (G ) and | Γ (u , v) |≥ 1 ) If D is minimal common neighbourhood dominating set, then V − D dominating set. Proof. Let D be a minimal common neighbourhood dominating set of G . Suppose V − D is not a dominating set.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

550

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

Then there exists a vertex u in D such that u is not dominated by any vertex in V − D . Then u is dominated by at least one vertex v in D − {u} and | Γ (u , v) |≥ 1 . Thus is a common neighbourhood D − {u} dominating set of G , which contradicts the minimal common neighbourhood dominating of D . Thus every vertex in D is adjacent with at least one vertex in V − D . Hence V − D is a dominating set. Corollary 2.10. Let G be a primitive strongly regular graph with the parameters (n, k , λ , µ ) such that λ > 0 . If D is a minimal common neighbourhood dominating set, then V − D is a dominating set. Definition 2.11. Let u ∈ V . The CNneighbourhood of u denoted by N cn (u ) is defined as Ncn (u) = {v ∈ N(u) :| Γ(u, v) |≥ 1}. The cardinality of N cn (u ) is denoted by d cn (u ) G , and N cn [u ] = N cn (u ) ∪ {u} . The in maximum and minimum common neighbourhood degree of a point in G are denoted respectively by ∆ cn (G ) and δ cn (G ) . That is ∆ cn (G ) = max u∈V | N cn (u ) | , δ cn (G ) = min u∈V | N cn (u ) | . Theorem 2.12. For any graph

[ 1+ ∆ n (G ) ] ≤ γ cn (G ) cn

Proof. Let S be a γ cn -set. Each vertex can common neighbourhood dominate at most γ cn (G )(∆ cn (G ) + 1) vertices, so n =| N cn [ s ] |≤ γ cn (G )(∆ cn (G ) + 1) , hence n ≤ γ cn (G ) . Therefore 1+ ∆ cn ( G ) [ 1+ ∆ n (G ) ] ≤ γ cn (G ) . cn

Corollary 2.13. For any strongly regular graph with the parameters ( n, k , λ , µ ) n ≤ γ (G ) . cn 1+ λ Proposition 2.14. Let G be a graph, γ cn (G ) = p if and only if G is a triangle free. Proof. Let G be a graph with common neighbourhood dominating set D and γ cn (G ) = p . We claim that G is a triangle free. Let uvw be the three vertices form a triangle. Now clearly D − {v} is a common neighbourhood dominating set of G . Hence γ cn (G ) ≤ p − 1 , a contradiction. Conversely, Let G be triangle-free graph. Let D be a common neighbourhood dominating set of G with | D |≤ p − 1 and v ∈ V − D , since D is a common neighbourhood dominating set of G . Then there exists at least one vertex u ∈ D such that uv ∈ E (G ) and | N (u ) ∩ N (v ) |≥ 1 . Let w ∈ N (u ) ∩ N (v ) , then u , v, w form a triangle, a contradiction. Corollary 2.15. If then γ cn (G ) = p .

is bipartite graph,

Let X be a random variable on a probability space Ω , and let E[ X ] be the expectation of X . Then we know that if E[ X ] ≤ c for some constant c , there is an s ∈ Ω such that X ( s) ≤ c . Let X 1 , X 2 ,..., X n be random variables, and let X = c1 X 1 + ... + cn X n , where ci 's are constants. Linearity of expectation states that [ X ] = c1[ X 1 ] + ... + cn [ X n ] . Using this simple observation, we prove the following theorem.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

Theorem 2.16. Let G be a graph with order n and minimum common neighbors of every two adjacent vertices k ≥ 1 . Then G has a minimal common neighbourhood dominating set of size at most 1

{1 − ( 1+1k ) k + ( 1+1k )

1+ k k

}n .

= ∑ v∈V P(v ∈ T ) ≤ n(1 − p)1+ k

(1)

Using elementary calculus, we minimize the left side of (1) with respect to p . Then the minimum value of it is 1

1+ k

{1 − ( 1+1k ) k + ( 1+1k ) k }n .

Which is attained when

The set D ∪T is clearly a common neighbourhood dominating set of G whose cardinality is at most 1+ k

Therefore, we have E[| D | + | T |] ≤ np + n(1 − p )1+k

1+ k

| D | + | T |≤ {1 − ( 1+1k ) k + ( 1+1k ) k }n .

{1 − ( 1+1k ) k + ( 1+1k ) k }n .

Proof. Fix p with 0 < p < 1 . Let us select, randomly and independently, each vertex of V = V (G ) with probability p . Let D be the random set of all vertices selected, and let T be the random set of all vertices not in D that do not have any common neighbors in D . Then the expectation E[| D |] of the random variable | D | is E[| D |] = np since | D | has a binomial distribution with parameters n and p . To find E[| T |] , we let | T |= ∑v∈V χ v , where χ v = 1 if v ∈ T and χ v = 0 otherwise. Note that P(v∈ T)= P (v and its common neighbors are not in D)= (1 – p)1+common neighbors of v ≤ (1 – p)1+k. for each v∈ V. Thus, we have E[| T |] = E[∑ v∈V χ v ] = ∑ v∈V [ χ v ]

551

p = 1 − ( 1+1k ) k .

This means that there is at least one choice of D such that

Definition 2.17. A subset S of V is called a common neighbourhood independent set (CN-independent set), if for every u ∈ S , v ∉ N nc (u ) for all v ∈ S − {u} . It is clear that every independent set is CNindependent set. A CN-independent set S is called maximal if any vertex set properly containing S is not CN-independent set, The maximum cardinality of CNindependent set is denoted by β cn , and the lower CN-independence number icn is the minimum cardinality of the CN-maximal independent set. Theorem .2.18. Let S be a maximal CNindependent set. Then S is minimal CNdominating set. Proof. Let S be a maximal CNindependent set. Let u ∈ V − S . If u ∉ N cn (v) for every v ∈ S , Then S ∪ {u} is CNindependent set, a contradiction to the maximality of S . Therefore u ∈ N cn (v) for some v ∈ S . Therefore S is CNdominating set. Since for any u ∈ S , u ∉ N cn (v) for every v ∈ S − {u} either N (u ) ∩ S = φ or | Γ(u, v) |= 0 for all v ∈ N (u ) ∩ S . Therefore set.

is minimal CN-dominating

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

552

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

Proposition 2.19. For any graph γ cn (G ) ≤ icn (G ) ≤ βcn(G ) ≤ Γcn ((G ) .

3. SOME GRAPHS RELATED TO THE COMMON NEIGHBOURHOOD Definition 3.1. Let G = (V , E ) be a graph. The neighbourhood graph G cn of G is defined as the graph with vertex set V (G ) and two distinct vertices u, v are adjacent if and only if they have at least one common neighbours. Theorem 3.2.

(ii) ( K r , m ) cn ≅ K r ∪ K m . (iii) (W p ) cn ≅ ( K p ) cn ≅ K p . (iv) if p ≥ 5 , then if p is odd; C p , (C p )cn ≅  C 2p ∪ C 2p , if p is even.

then µ ≥ 1 and λ = 0 and by the definition of common neighbourhood graph, any two vertices in (G) cn are adjacent if they are not adjacent in G . Hence (G) is the G cn complement of G which is strongly regular graph with parameters (n, n − 2k − 2 + µ, n − 2k + λ) . (ii) it is clear that any two vertices in (G) cn has either λ or common µ neighbourhoods. Therefore any two vertices in (G) cn are adjacent. Hence (G )cn = K n . be primitive Corollary 3.4. Let G strongly regular graph with order p and domination number γ . Then  p, if G has no triangles;

γ cn (G ) =  γ , otherwise.

Theorem

3.5.

For

any

graph

γ ((G ) ) ≤ γ cn (G ) . cn

We omit the proof of the above proposition because it is easy to prove all the parts of the proposition directly from the definition of the common neighbourhood graph.

Proof. It is clear that any common neighbourhood dominating set of G is dominating set of (G)) cn . Hence

Theorem 3.3. Let G be primitive strongly regular graph with parameters (v, k , λ , µ ) then (i) (G) cn is strongly regular graph with parameters (v, v − 2k − 2 + µ , v − 2k + λ ) if G is strongly regular graph without triangles. (ii) (G )cn = K v otherwise.

Theorem 3.6. Let G be a graph such that the set which has common neighbourhood points with u is contained in N (u ) for all

Proof. (i) It is clear since G be primitive strongly regular graph without triangles,

γ ((G )cn ) ≤ γ cn (G ) .

u ∈ V . Then γ (G cn ) = γ cn (G ) . Proof. Let D be any γ − set of G cn . Then for any u ∈ V − D there exist v ∈ D such that u and v are adjacent in G cn . That is u and v has common neighbourhood in G , and by hypothesis,

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580))

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

u ∈ N (v) . Therefore

is common neighbourhood dominating set of G. cn Therefore γ cn (G ) ≤| D |= γ (G ) , but

γ ((G)cn ) ≤ γ cn (G) . Hence γ (Gcn ) = γ cn (G) . Definition 3.7. Let G be a given graph. Let H be the graph constructed from G as follows: V ( H ) = V (G ) , two points u and v are adjacent in H if and only if u and v are adjacent and have common neighbourhood in G . H is called the adjacent inherent common neighbourhood graph of G or common neighbourhood associate of G and is denoted by cn(G ) . It is clear that E (cn(G )) ⊆ E (G ) , because if e = uv ∈ E (cn(G )) , then u and v are adjacent and have common neighbourhood in G . Therefore e ∈ E (G ) . An edge e = uv ∈ E (G ) is said to be common neighbourhood edge (CN-edge) edge) if | Γ(u,v) |≥1. Observation 3.8. Let G be a graph. If every edge in G is common neighbourhood edge, then γ (cn(G )) = γ (G ) .

553

Proof. Let E cn (G ) be the set of all common neighbourhood edges of G . Then that is clear E cn (G ) = E (cn(G )) . Hence cn(G ) = G , that is γ (cn(G )) = γ (G ) . Observation 3.9. If G is a graph without common neighbourhood edge, then cn(G ) is totally disconnected. Theorem 3.10. Let G is primitive strongly regular graphs with triangles. Then cn(G ) = cn(G ) . is primitive strongly Proof. since G regular graph with triangles, then it is clear that every edge is common neighbourhood, that is cn(G ) = G . Similarly cn(G ) = G . Hence cn(G ) = cn(G ) . It is not true in general for all graphs if every very edge is CNCN edge in G then in G every edge is CNCN edge.

Example 3.11. Consider in a graph G every edge is common neighbourhood as in Figure 2a

It is clear that in G the edge ef is not CN-edge (see figure 2b) Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580))

554

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

4. TOTAL CN-DOMINATING SETS, CN-COVERING SETS AND CN-NEIGHBOURHOOD SETS Definition 4.1. A subset D ∈ V is said to be total common neighbourhood dominating set (total CN-dominating set) if for all x ∈ V , there exist y ∈ D such that y is adjacent to x and y has common neighbourhood with x . If G has no common neighborhood isolated points, then V is total common neighbourhood set. The minimum cardinality of a total common neighbourhood dominating set in a graph G is called the total common neighbourhood domination number of G denoted by γ tcn . It is convention to take γ tcn = ∞ if G has a common neighbourhood isolated point.

Proposition 4.2. (i) γ tcn ( K p ) = γ tcn (W p ) = 2 (ii) γ tcn ( Pp ) = γ tcn (C p ) = γ tcn ( K r , m ) = ∞

Proof. The proof of the proposition is very easy we can get it directly from the definition of the total common neighbourhood dominating set. Definition 4.3. Let G = (V , E ) . A subset S of V is called Common neighbourhood vertex covering (CN-vertex covering) of G if for CN-edge e = uv either u ∈ S or v ∈ S . The minimum cordiality of CNvertex covering of G is called the CNcovering number of G and denoted by α cn (G) .

Proposition 4.4. (i) If G is a graph has no CN-edge, then α cn (G) = 0 (ii) If G is strongly regular graph with parameters (n, k , λ , µ ) such that λ > 0 , then α cn (G ) = α (G) . (iii) α cn (W p ) = [ 2 ] + 1 p

(iv) Let G be any graph has no common neighbourhood isolates. Then every common neighbourhood vertex cover is common neighbourhood dominating set of G , i.e,. γ cn (G) ≤ α cn (G ) ≤ α (G) .

Theorem 4.5. Let G be a graph of order n . Then

α cn (G ) + β cn (G ) = n

Proof. Let S be β cn -set in G , suppose e = uv be any CN-edge, then either u or v are in V − S , that is V − S is common G. neighbourhood vertex cover of Therefore | V − S |≤ α cn (G) . That is

n ≤ α cn (G ) + β cn (G ) . Similarly if we suppose that S be α cn -set in G , and e = uv be any CN-edge, so one of the points u and v must belongs to S , that

is V −S is common neighbourhood independent set in G . Therefore | V − S |≥ β cn (G ) , that is n ≥ α cn (G) + βcn (G) . Hence α cn (G) + βcn (G) = n .

Definition 4.6 A set of CN-edges is said to be independent if no two edges in that set have a common point. The common

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

neighbourhood edge independence number denoted by β cn1 (G) is the maximum cardinality of an independent set of CN CNedges. From the definition ition it is easy to see that for any graph G . β cn1 (G ) = β cn1 (cnG) .

Definition 4.7. Let G be a graph. Let S be a subset of V (G ) . The common neighbourhood subgraph ind induced by S is the subgraph whose vertex set is S and two points of S are adjacent if and only if they are adjacent and they have common neighbourhood in G . Definition 4.8. Let G = (V , E ) be a graph.

Proposition 4.10. (i) The CN-neighbourhood neighbourhood number of any graph G of order n without triangles is n . (ii) ηcn (W p ) = 1

555

E ncn be the set of non-common non Let neighbourhood edges of G . A subset S of V (G ) is a common neighbourhood set of G if G − E ncn = ∪v∈S ((〈 N cn [v]〉)cn ) , where (〈 Ncn[v]〉)cn is the common neighbourhood subgraph of G induced by N cn [v] . The common neighbourhood number ηcn of G is the minimum cardinality of a common neighbourhood set of G .

Example 4.9: Let G be a graph as in Figure 3a, then 〈Ncn[6]〉= (〈Ncn[6]〉)cn as in Figure 3b.

Theorem 4.11. If every edge in a graph G is CN-edge, then ηcn (G) = η (G) . Proof. If every edge in G is CN-edge CN then G = (cnG) , then ηcn (G) = η (G) .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580))

556

Anwar Alwardi, et al., J. Comp. & Math. Sci. Vol.2 (3), 547-556 (2011)

The converse of previous theorem is not true for example consider the graph H obtained by joining any vertex of K p with extra vertex, here ηcn ( H ) = η ( H ) = 2 , but H has edge which is not CN-edge.

Theorem 4.12. For any graph G , γ (G ) ≤ η (G) ≤ γ cn (G) . In the final we state the following Problems:

open

Problem 1. Characterize graph G for which γ (G ) = η (G) = γ cn (G) . Problem 2. If a , b , c be four integers such that a ≤ b ≤ c . Then there exists a graph G with γ (G ) = a , η (G ) = b and γ cn (G) = c . REFERENCES 1. S. Arumugam and C. Sivagnanam, Neighborhood connected and neighborhood total domination in graphs. Proc. Int. Conf. on Disc. Math., 2334 B. Chaluvaraju, V. Lokesha and C. Nandeesh Kumar Mysore, 45-51 (2008).

2. B. Chaluvaraju, Some parameters on neighborhood number of a graph, Electronic Notes of Discrete Mathematics, Elsevier, 33, 139–146 (2009). 3. F. Harary, Graph theory, AddisonWesley, Reading Mass (1969). 4. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc., New York (1998). 5. S. M. Hedetneimi, S. T. Hedetneimi, R. C. Laskar, L. Markus and P. J. Slater. Disjoint dominating sets in graphs. Proc. Int. Conf. on Disc. Math., IMI-IISc, Bangalore, 88 - 101(2006). 6. V. R. Kulli and S. C. Sigarkanti, Further results on the neighborhood number of a graph. Indian J. Pure and Appl. Math.23 (8), 575 -577 (1992). 7. E. Sampathkumar and P. S. Neeralagi, The neighborhood number of a graph, Indian J. Pure and Appl. Math.16 (2), 126 – 132 (1985). 8. H. B. Walikar, B. D. Acharya and E. Sampathkumar, Recent developmentsin the theory of domination in graphs, Mehta Research instutute, Alahabad, MRI Lecture Notes in Math. 1 (1979).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)