SPLIT BLOCK SUBDIVISION DOMINATION IN GRAPHS

International Journal of Technical Research and Applications e-ISSN: 2320-8163,
www.ijtra.com Volume 2, Issue 5 (Sep-Oct 2014), PP. 82-86
82 | P a g e
SPLIT BLOCK SUBDIVISION DOMINATION IN
GRAPHS
M.H. Muddebihal 1, P.Shekanna2, Shabbir Ahmed3
Department of Mathematics,
Gulbarga University,
Gulbaarga-585106.
1
mhmuddebihal@yahoo.co.in, 2
shaikshavali71@gmail.com, 3
glbhyb09@rediffmail.com
Abstract: A dominating set is a split dominating
set in . If the induced subgraph is
disconnected in The split domination number of
is denoted by , is the minimum cardinality of
a split dominating set in . In this paper, some results on
were obtained in terms of vertices, blocks, and other
different parameters of but not members of
Further, we develop its relationship with other different
domination parameters of
Key words: Block graph, Subdivision block graph, split
domination number.
[I] INTRODUCTION
All graphs considered here are simple, finite, nontrivial,
undirected and connected. As usual denote the
number of vertices, edges and blocks of a graph
respectively. In this paper, for any undefined term or notation
can be found in F. Harary [3] and G .Chartrand and PingZhang
[2]. The study of domination in graphs was begin by O.Ore [5]
and C.Berge [1].
As usual, The minimum degree and maximum degree of a
graph are denoted by respectively. A
vertex cover of a graph is a set of vertices that covers all the
edges of The vertex covering number is a minimum
cardinality of a vertex cover in The vertex independence
number is the maximum cardinality of an independent
set of vertices. A edge cover of is a set of edges that covers
all the vertices. The edge covering number of is
minimum cardinality of a edge cover. The edge independence
number of a graph is the minimum cardinality of an
independent set of edges.
A set of vertices is a dominating set. If every
vertex in is adjacent to some vertex in .The
Domination number of is the minimum cardinality of
a dominating set in .
A dominating set of a graph is a split dominating set
if the induced subgraph is disconnected. The split
domination number of a graph is the minimum
cardinality of a split dominating set .This concept was
introduced by A dominating set of is a cototal
dominating set if the induced subgraph has no
isolated vertices. The cototal domination number of
is the minimum cardinality of a cototal dominating set. See
[4]
The following figure illustrate the formation of
of a graph
The domination of split subdivision block graph is denoted
by . In this paper, some results on where
obtained in terms of vertices, blocks and other parameters of
We need the following Theorems for our further results:
[II] MAIN RESULTS
Theorem A [4]: A split dominating set of is minimal for
each vertex one of the following condition holds.
There exists a vertex such that
is an isolated vertex in
is connected.
Theorem B [4]: For any graph .
Now we consider the upper bound on in terms of
blocks in
Theorem 2.1: For any graph with
then .

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Proof: For any graph with , a split domination
does not exists. Hence we required blocks. Let
be the number of blocks of
and be the vertices in
with corresponding to the blocks of Also
be the set of vertices in
Let ,
1 be a set of cut vertices. Again consider a
subset such
that . Let
,1 which
are not cut vertices such that
is a dominating set
.Clearly is disconnected graph.
Then is a Hence
which gives
In the following Theorem, we obtain an upper bound for
in terms of vertices added to
Theorem 2.2: For any connected graph with
blocks, then where is the number of vertices
added to
Proof: For any nontrivial connected graph . If the graph
has Then by the definition, split domination set
does not exists. Hence Let
be the blocks of and
be the vertices in
which corresponds to the blocks of .Now we consider the
following cases.
Case1: Suppose each block of is an edge.
Then . Let
be the set of vertices of Now
consider , is a set of
cut vertices in
are adjacent to end vertices of
Again there exists a subset of with the
property where is
adjacent to atleast one vertex of is a
disconnected graph . Hence is a
By Theorem 1,
Case2: Suppose each block of is a complete graph
with . Again we consider the sub cases of
case 2.
Subcase2.1: Assume Then
and
where
is an isolates. Hence which
gives .
Sub case 2.2: Assume every block of ) is .
Let then
and
=
where
is an isolate. Hence
which gives
We establish an upper bound involving the Maximum degree
and the vertices of for split block sub division
domination in graphs.
Theorem 2.3: For any graph
then .
Proof: For split domination, We consider the graphs with
the property Let
be the blocks of and
be the vertices in
corresponding to the blocks of Let
be the vertices in . Let
be a By Theorem A, each vertex
there exist a vertex is a split
dominating set in Thus
Since by
Theorem B which gives
The following lower bound relationship is between split
domination in and vertex covering number in

84 | P a g e
Theorem 2.4: For any graph with ,then
, where is a vertex covering
number of
Proof: We consider only those graphs which are not
Let be the blocks of
which correspondes to the set
be the vertices in B(G). Let
be the vertices in
Again
such that
, and
. Hence is disconnected, which
gives . Now
and each edge in is adjacent to atleast one vertex in
Clearly . Hence
which gives .
The following result gives a upper bound for in terms
of domination and end blocks in
and then
.
Proof: Suppose graph is a block .Then by definition, the
split domination does not exists. Now assume is a graph
with at least two blocks. Let
be the set of blocks in and
be the vertices in which corresponds to the blocks of .
Now be the vertices
in . Suppose D is a of ,
whose vertex set is . Note that
at least one . More over, any component of is of
size atleast two. Thus is minimal which gives
. Again
be the vertices in and
Every
vertex of is adjacent to at least one vertex of
Suppose there exists a vertex such that every vertex of
is not adjacent to at least one vertex
Thus . Hence
which gives .
A relationship between the split domination in and
independence number of a graph is established in the
following theorem.
Theorem2.6: For any connected graph with
then is the
independence number of .
Proof: By the definition of split domination, Let
be the blocks of which
corresponds to the vertices of the set
in . Let
be the vertices in
such that .Let be the
set of vertices in e have the following cases.
Case1: Suppose is a tree. Let
are cut vertices in
Again
and ,were . Then we consider
,
where
with the property
and is a set of all end vertices
in . Again where every is an
isolates.Thus
Case 2: Suppose is not a tree. Again we consider sub
cases of case 2
Subcases2.1: Assume is a block. Then
in
Thus the number of isolates in
Hence . One
can see that for the as in case , We have
which gives .
Sub case 2.2: Assume has atleast two blocks.Then as in
subcase 2.1,we have .
The next result gives a lower bound on in terms of
the diameter of
Theorem 2.7: For any graph with blocks
,then .

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Proof : Suppose be the
blocks of ,Then be the
corresponding block vertices in B(G). Suppose
be the set of edges which
constitutes the diameteral path in .Let where
Suppose are non end blocks
in which gives cut vertices in and
Suppose be the vertices in
where
such that are cut
vertices in Since they are non end blocks in
. Then is a . Clearly
Suppose is cyclic then there exists atleast one block
which contains a block diametrical path of length atleast two.
In the block as a singleton and if
atmost two elements of diameter of then
gives .
Suppose is acyclic then each edge of is a block of
Now , where gives
. Clearly we
have .
The following result is a relationship between ,
domination and vertices of .
.
Proof: Suppose the graph has one block, then split
domination does not exists. Hence
Suppose be the blocks of
Then be the
corresponding block vertices in Let
be the set of vertices in .
Also
and is adjacent to atleast one vertex of
be
the set of vertices in Now where
are non end blocks in .
Then we have which corresponds to the elements of
such that forms a minimal dominating set of
Since each element of is a cut vertex, then
Further which
gives .
Next, the following upper bound for split domination in
is interms of edge covering number of .
Theorem2.9: For any connected ( graph with
where
is the edge covering number.
Proof: For any non trivial connected graph with
then by definition of split domination, the split
domination set does not exists. Hence .
Let be the blocks of
which correspondes to the set
be the vertices in Let
be the vertices in
such that We have the following cases.
Case 1: Suppose each block is an edge in .Then
where is the set of end
edges, If every cut vertex of is adjacent with an end
vertex. Then Then
.Otherwise .
Let then there exist
atleast one cut vertices in Let
which are non cut vertices in
Again and . The
is a split dominating set.
Hence . Since
has more than one component.
Hence which gives
Case2: Suppose has atleast one block which is not an edge.
Let , and
be the set of cut vertices such that . Again
, be the set of cut
vertices in such that
, and
Hence is disconnected
, which gives As in case 1 ,
will increase. Hence
which gives .

86 | P a g e
The following lower bound for split domination in is
interms of edge independence number in
Proof: By the definition of Split domination, we need
We have the following cases.
Case 1: Suppose each block in is an edge. Let
be the set of edges in
Also be a set of alternative edges in
Then
Consider be the vertices in
, again be the cut
vertices which are adjacent to at least one vertex of and
are the end vertices
in is
disconnected. Then
which gives
Case2: Suppose there exists at least one block which is not an
edge. Let be the set of edges in
Again is the set of alternative
edges in which gives
Suppose be the vertices of
Then where is a set of cut vertices
and is a set of non cut vertices. Now we consider
such that
has more than one component.
Hence is a and
which gives
In the following theorem, we expressed the lower bound for
in terms of cut vertices of
then where is the cut
vertices in .
Proof: Suppose graph is a block. Then by the definition, of
split domination, consider the following cases.
Case 1: Suppose each block of is an edge. Then we
consider be the cut vertices in
be the vertices in
and are cut vertices in
is adjacent to at least one vertex in
Then gives disconnected graph. Thus
gives .
Case 2: Suppose each block in is not an edge. Let
be the cut vertices in
Again
are the non cut vertices in
.Further we consider such that
where is disconnected.
Clearly which gives
Finally, the following result gives an lower bound on
in terms of .
Theorem 2.12: For any nontrivial tree with
.
Proof: We consider only those graphs which are not
Let H
, be a subset
of which are end vertices in Let
with
such that and
has no isolates,
then . Let
be the vertices in consider
be the set of
all vertices of with
the property , is a set of all
end vertices in
gives minimum split domination in
. Clearly which
gives .
REFERENCES
[I] C Berge, Theory of graphs and its applications, Methuen,
London, (1962).
[II] G. Chartrand and Ping Zhang, “Introduction to graph Theory”,
Newyork (2006).
[III] F.Harary, Graph Theory, Adison Wesley, Reading Mass (1972).
[IV] V.R.Kulli, Theory of domination in Graphs, Vishwa international
Publications, Gulbarga, India. (2010).
[V] O.Ore, Theory of graphs, Amer. Math. soc., Colloq. Publ., 38
Providence, (1962).

SPLIT BLOCK SUBDIVISION DOMINATION IN GRAPHS

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SPLIT BLOCK SUBDIVISION DOMINATION IN GRAPHS