A Note on Non Split Locating Equitable Domination

Integrated Intelligent Research (IIR)

International Journal of Data Mining Techniques and Applications Volume: 01 Issue: 02 December 2012 Page No.26-28 ISSN: 2278-2419

A Note on Non Split Locating Equitable Domination P.Sumathi 1 G.Alarmelumangai 2 Head & Associate Professor in Mathematics, C.K.N college for Men,1 Anna Nagar, Chennai, India 2 Lecturer in Mathematics, E.M.G Yadava women’s college, Madurai, India.

Email: 1 sumathipaul@yahoo.co.in2 alarmelu.mangai@yahoo.com 1

Abstract - Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DV 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 (G) of G is the minimum cardinality of a dominating set. A dominating set D is called a non split locating equitable dominating set if for any two vertices u,wV-D, N(u)D  N(w)D, N(u)D=N(w)D and the induced sub graph V-D is connected.The minimum cardinality of a non split locating equitable dominating set is called the non split locating equitable domination number of G and is denoted by nsle(G).

In this paper, bounds for nsle(G) and exact values for some particular classes of graphs were found.

nsle (G). A set D is said to be a nsle – set if D is a minimum non split locating equitable dominating set.

Keywords-Domination number

A. Proposition

Number,non

Split

II. CHARACTERIZATION OF NON SPLIT LOCATING EQUITABLE DOMINATING SET. Observation 1.

For any connected graph G,  (G)  nsle (G).

2.

For any connected spanning subgraph H of G , nsle ( H )  nsle ( G ).

Example: (i) nsle ( C4 ) = nsle ( P4 )=2. (ii) nsle ( K4 ) = 3, nsle ( C4 ) = 2.

domination

Pendant vertices are members of every nsle-set.

Proof. Let v be vertex in G such that deg(v) =1 and let D be a nsle-set. If vV-D, then a vertex adjacent to v must be in D and hence V-D is disconnected, which is a contradiction.

I. INTRODUCTION For notation and graph theory terminology [2] is followed.

Specifically, let G=(V,E) be a simple, undirected, finite nontrivial graph with vertex set V and edge set E. For a vertex vV, the open neighborhood of v is the set NG(v)={u /uv  E}, NG(v) can be written as N(v) and the closed neighborhood of v is the set NG[v]= N(v){v} , NG[v] can be written as N[v] . The degree of a vertex v is the number of edges incident with v in G, i.e, d(v) =N(v). The maximum , minimum degree among the vertices of G is denoted by (G) ,(G) respectively.

If deg v = 0,then v is called an isolated vertex of G. If deg v = 1 , then v is called a pendant vertex of G.As usual Kn, Cn, Pn and K1,n-1 denote the complete graph, the cycle, the path and the star on n vertices, respectively. The distance dG (u,v) or d (u,v) between two vertices u and v in a graph G, is the length of a shortest path connecting u and v. The diameter of a connected graph G is the maximum distance between two vertices of G it is denoted by diam(G) . A non empty set DV 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 (G) of G is the minimum cardinality of a dominating set. A dominating set D of a graph G = (V,E) is a non split dominating set if the induced subgraph V-D is connected .

The non split domination number ns(G) of a graph G is the minimum cardinality of a non split dominating set. A set D is said be a ns-set if D is a minimum non split dominating set. A dominating set D is said to be non split locating equitable dominating set if for any two vertices u,wV-D, N(u)D  N(w)D, N(u)D=N(w)D and the induced sub graph V-D is connected . The minimum cardinality of a non split locating equitable dominating set is called the non split locating equitable domination number of G and is denoted by

B.

Proposition

nsle (G)  e where e is the number of pendant vertices.

Proof. Since every pendant vertex is a member of each non split locating equitable dominating set.

III. BOUNDS OF NON SPLIT LOCATING EQUITABLE DOMINATION NUMBER. n ber .

Observation .- For any connected graph G with n  2, nsle (G)  n -1.

This bound is sharp for Kn. Theorem- For any connected (n,m) graph G with (G)  2, nsle (G)  3n- 2m-2.

Proof. Let D be a nsle –set of G and let t be the number of edges in G having one vertex in D and the other in V-D. Number of vertices in <V-D> is n-nsle (G) and minimum number of edges in <V-D> is n-nsle (G) -1 . Hence  viD deg (vi) + t. Since V-D= n-nsle (G) , there are atleast n-nsle (G) edges from VD to D. Also deg(vi )  (G) Therefore 26

Integrated Intelligent Research (IIR)

International Journal of Data Mining Techniques and Applications Volume: 01 Issue: 02 December 2012 Page No.26-28 ISSN: 2278-2419 equitable dominating set. Therefore G is the graph obtained from a complete graph by attaching pentant edges at atleast one of the vertices.

Case (ii)  (G) = 2 . V-D ={w1,w2 }. Let w be vertex of degree  3 in G and w V – D and w = w’. Let each vertex of D be adjacent to both w1 and w2. If <D> is complete, then G is complete. Assume < D > is not complete. Then there exists atleast one pair of non-adjacent vertices in D, say u,vD and V – { u, v, w1 } is a non split locating equitable dominating set of G containg (n-3 ) vertices, which is a contradiction.Therefore there exists a vertex in D which is adjacent to exactly one of w1 and w2 and again we get a non split locating equitable dominating set having (n-3) vertices and hence wD. Since deg (w )3, there exists atmost one vertex, say vD, adjacent to w. Then either V – { v,w,w1 } or V- {v,w, w2 } will be a non split locating equitable dominating set of G. Therefore there exists no vertex of degree 3 in G and hence each vertex in G of degree 2 and G is a cycle.

Case (iii)  (G) = 1, Let u,v be non adjacent vertices in < D >.

Then either V – {u,v w1 } or V- { u,v,w2} will be a non split locating equitable dominating set, which is a contradiction.

Therefore , < D {w1,w2}> is path. Hence G  Pn..

(G) nsle (G) + n-nsle (G)  2 [ m –(n--nsle (G) -1]. But (G)  2, Hence, 2 nsle (G) + n-nsle (G)  2 [ m –n + nsle (G) +1]. Thus, 3n- 2m -2  nsle (G) Therefore nsle (G)  3n- 2m-2.

Remark. This bound is attained if G  Cn, n  3.

Theorem 3.3. Let G be a connected graph and (G) =1. Then, nsle (G)  3n- 2m-e-2, where e is the number of pendant vertices.

Proof. Let D be a nsle - set of G , such that D= nsle (G) and let t be the number of edges in G having one vertex in D and the other in V-D. As in Theorem 3.2, 2 [ m – ( n - nsle (G) – 1] =  viD deg ( vi ) + t  e + 2 ( nsle (G) – e ) + n-nsle (G) Hence , nsle (G)  3n- 2m-e-2.

Remark. This bound is attained if G  Pn, n  3.

Corollary 3.4 . If G is a connected k-regular graph ( k > 3 ) with n vertices, then nsle (G)  (n ( k -3) +2 ) / ( k – 3 ).

Theorem 3.8. For a connected graph G, nsle (G) = e if and only if each vertex of degree atleast 2 is a support, where e is the number of pendant vertices in G. Proof. Assume each vertex of degree atleast 2 is a support. If S is the set of all pendant vertices in G , then S is a dominating set in G and since <V-S> is connected, S is a nsle – set of G. Therefore nsle (G)  e By proposition 2.3 theorem follows.

Conversely, let u be a vertex in G such that deg (u)  2 and Let D be a nsle – set of G. If u is not a support of G, then u is not adjacent to any of the vertices in D, which is a contradiction.

Corollary 3.5. If (G) > 3, then nsle (G)  ( 2m- 3n + 2) / ((G) – 3).

Theorem 3.6. Let G be a connected graph with n  2, nsle (G) = n-1 if and only if G is a star and G is a complete graph on n vertices.

Proof. If G  K1, n-1 then the set of all pendant vertices of ,. K1, n-1 forms a minimal non split locating equitable dominating set for G. Hence nsle (G) = n-1.

Conversely assume nsle (G) = n-1. Then there exists a non split locating equitable dominating set D containing n-1 vertices. Let V- D = {v}. Since D is a dominating set of G, v is adjacent to atleast one of the vertices in D,say u . If u is adjacent to any of the vertices in D, then the vertex u must be in V-D. Since D is minimal, u is adjacent to none of the vertices in D. Hence G  K1, n-1 and Kn.

Theorem 3.9. If G is a connected graph which is not a star, then nsle (G)  n-2.

Proof. Since G is not a star, there exists two adjacent cut vertices u and v with deg (u), deg(v)  2. Then V- {u,v} is a non split locating equitable dominating set of G. Hence nsle (G)  n-2.

Theorem 3.7. Let G bea connected graph nsle (G) = n-2 if and only if G is isomorphic to one of the following graphs. Cn, Pn or G is the graph obtained from a complete graph by attaching pendant edges at atmost one of the vertices of the complete graph and atmost at n-1 vertices.

Proof. For all graphs given in the theorem, nsle (G) = n-2.

Conversely, let G be a connected graph for which nsle (G) = n-2. and let D be a non split locating equitable dominating set of G such that D= n – 2 and V- D = {w1,w2 } and < V – D >  K2.

Case (i) By Proposition on 2.2, all vertices of degree 1 are in D and any vertex of degree 1 in D are adjacent to atmost one vertex in V-D since < V – D >  K2. Also each vertex in V- D is adjacent to atmost on vertex in D. Let D’ = D – { pendant vertices }. Then {w1,w2 }  D’ will be a complete graph. Otherwise, there exists a vertex u D’ , such that u is not adjacent to atleast one of the vertices of D’– {u} and hence D-{u} is a non split locating

Theorem 3.10. Let T be a tree with n vertices which is not a star. Then nsle (G) = n-2 if and only if T is a path or T is obtained from a path by attaching pendant edges at atleast one of the end vertices.

Proof. Let T be a tree which is not a star. It can be easily verified that for all trees stated in the theorem nsle (G) = n-2.

Conversely, Assume nsle (G) = n-2. Let D be a nsle- set containing n-2 vertices and let V-D= {w1,w2} and <V-D>  K2. Since T is a tree, each vertex in D is adjacent to atmost one vertex in V-D. Since D is a dominating set, each vertex in V-D is adjacent to atleast one vertex in D. (i) If <D> is independent, then T  double star. (ii) Assume <D> is not independent. Then there exists a vertex u <D> such that deg(u) 1 in <D>. Also either Nj(u)= 1, 1 j  diam ( T ) – 3 or if Nj(u) 2 for some j , 27

Integrated Intelligent Research (IIR)

International Journal of Data Mining Techniques and Applications Volume: 01 Issue: 02 December 2012 Page No.26-28 ISSN: 2278-2419

j  diam ( T ) – 4, Then < Nj(u)> in D is independent, since otherwise D- {u} is a non split locating equitable dominating set of T. In the first case T is a path. In the second case, T is a tree obtained from a path by attaching pendant edges at atleast one of the end vertices of the path.

 (G) = n-2 and nsle (G) +  (G) < 2n-3 implies n=4.Hence, G  C4. Let G be the graph obtained from a cycle by attaching pendant edges at atleast one of the vertices of the cycle. Let n be the number of vertices in the cycle and e be the maximum number of pendant edges attached.

Hence  (G) = n – 5 +e, Therefore nsle (G) +  (G) < 2n-3.

Let G  Pn, nsle (G) +  (G) < 2n-3 implies n =4 . Hence G  P4. Let G be a graph obtained from a path by attaching pendant edges at atleast one of the end vertices. Let e be the number of pendant edges attacted , e  2. Hence, nsle (G) +  (G) < 2n-3 .

IV. RELATION BETWEEN NON SPLIT LOCATING EQUITABLE DOMINATION NUMBERAND OTHER PARAMETER Theorem 4.1. For any tree with n vertices, c ( T) + nsle ( T )  n.

Proof. If D1 is the set of all cut vertices of T with D1= n1, then c ( T) = n1. If D2 is the set of all pendant vertices of T with D2= n2, then nsle ( T )  n2. But V ( T1 )= n1 + n2 implies that c ( T) + nsle ( T )  n.

By theorem 3.6 , equality holds if and only if each vertex of degree atleast 2 is a support.

Proposition 4.5. For any connected (n,m) graph G, nsle ( G ) +  (G)  2n-2.

REFERENCS [1] J. Cyman, The outer-connected domination number of a graph.

Australasian J. Comb. 38 (2007), 35-46.

[2] T.W. Haynes, S.T.Hedetniemi, and P.J. Slater, Fundamentals of domination Graphs, Marcel Dekker, New York, 1998.

[3] V.R. Kulli and B. Janakiram, The non split domination number of a graph.

India J. Pure Appl.math; 31(2000)545-550.

[4] P. Sumathi , G.Alarmelumangai, Non split locating equitable domination, Proceedings.Of the International Conference on Mathematics in Engineering & Business Management, March 9-10,2012.ISBN: 978-81-8286-015-5.

[5] Hongxing Jiang, Erfang shan, outer-connected domination in graphs, Utilitas Mathematics 81(2010).

Proof . For any graph with n vertices,  (G)  n-1.

By observation 3.1, the proposistion follows.

Proposition 4.6 For any connected (n,m) graph G, nsle ( G ) +  (G) = 2n-2 if and only if G  K1,n-1 or Kn. Proof. When G  K1,n-1 or Kn., nsle (G) +  (G) = 2n-2.

Conversely, nsle (G) +  (G) = 2n-2. is possible if nsle (G) = n-1 and  (G) = n-1.

But, nsle ( G ) = n-1 is possible if and only if G is a star and G is a complete graph.

Theorem 4.2. For any connected (n,m) graph G , nsle (G) +  (G) > 2n-3 and nsle (G) +  (G) < 2n-3 if and only if G is one of the following. (i) C3,P3,Kn, or G is the graph obtained from a complete graph by attaching pendant edges at exactly one of the vertices of complete graph. (ii) C4, P4 or G is the graph obtained from a cycle or path by attaching pendant edges at exactly one of the vertices of cycle or path.

Proof. For the graphs given in the theorem, nsle (G) +  (G) > 2n-3 and nsle (G) +  (G) < 2n-3 .

Conversely, nsle (G) +  (G) > 2n-3 is possible if nsle (G) = n-1 and  (G) = n-1 and nsle (G) +  (G) < 2n-3 possible if . nsle (G) = n-2 and  (G) = n-2.

In the first case, nsle (G) = n-1 if and only if G is a star on n vertices .But for a star  (G) = n-1 and hence this case is possible.In the second case, nsle (G) = n-2 if and only if G is isomorphic to one of the following graphs. (a) Cn , Pn or G is obtained from a path by attaching pendant edges at atleast one of the end vertices.Let G  Cn, then . nsle (G) = n-2 and 28