This document discusses non split locating equitable domination in graphs. It begins with definitions of terms like domination number and non split locating equitable dominating set. It then presents several theorems that establish bounds on the non split locating equitable domination number of a graph based on its properties. These include bounds related to the number of vertices, minimum degree, number of pendant vertices, and whether the graph is regular or a tree. The document also characterizes the graphs that achieve equality in some of the bounds. In general, it analyzes the non split locating equitable domination number and relates it to other graph parameters.

1. 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
26
A Note on Non Split Locating Equitable Domination
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.
Keywords-Domination Number,non Split domination
number
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
nsle (G). A set D is said to be a nsle – set if D is a minimum non
split locating equitable dominating set.
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.
A. Proposition
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.
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 V-
D to D. Also deg(vi )  (G)
Therefore
P.Sumathi 1
G.Alarmelumangai 2
1
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

2. 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
27
(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 ).
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 3.7. 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
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..
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.
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 3.10. 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 ,

3. 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
28
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.
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.
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
 (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 .
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