 Open Access
 Authors : M. Reddappa , Dr. C. Jaya Subba Reddy
 Paper ID : IJERTV8IS100185
 Volume & Issue : Volume 08, Issue 10 (October 2019)
 Published (First Online): 25102019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Total Roman Domination In Special Type Interval Graph
M. Reddappa
Dept. of Mathematics, S.V.University, Research Scholar,
Tirupati517502, India..
Dr. C. Jaya Subba Reddy
Dept. of Mathematics, S. V. University, Assistant Professor,
Tirupati517502, India..
Abstract Interval graphs have drawn the attention of many researchers for over 40 years. They form a special class of graphs with many interesting properties and revealed their practical relevance for modeling problems arising in the real world. The theory of domination in graphs introduced by O. Ore [10] and C. Berge [1] has been ever green of graph theory
A Roman dominating function on a graph is a function satisfying the condition that every vertex for which is adjacent to at least
one vertex for which The weight of a Roman
today. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two
dominating function is the value
f (v) . The
vV
books by T. W. Haynes et.al. [12, 13].
Keywords Total domination number, Total Roman dominating function, Total Roman domination number, Interval family, Interval graph.
1. INTRODUCTION
Domination in graphs has been studied extensively in recent years and it is an important branch of Graph Theory. R.B.Allan and R.C. Laskar [11], E.J. Cockayne and S.T. Hedetniemi, [4] and many others have studied various domination parameters of graphs.
Let be a graph. A total dominating set of a graph G with no isolated vertices is a set S of vertices of G such that every vertex in V(G) is adjacent to at least one vertex in S.
The minimum cardinality of a total dominating set is called as total domination number and it is denoted by . A total dominating set of G of cardinality is called a .set.
Total domination in graphs was introduced by E. J. Cokayne et al. [6]. Total domination is now well studied in graph theory. The literature on the subject of total domination in graphs has been surveyed and detailed in the recent book by M.A. Henning et al. [9].
We consider finite graphs without loops and multiple edges.
minimum weight of a Roman dominating function on a graph is called as the Roman domination number of . It is denoted by . If = 2 then G is called a Roman graph.
Let and let be the ordered partition of induced by f where
for Then there
exists a 11 correspondence between the functions
and the ordered partition of
. Thus we write .
A function becomes a Roman dominating function if the set dominates .
Total Roman domination in graphs are studied by
Ahangar et al. [7]. A total Roman dominating function of a graph G with no isolated vertices, is a Roman dominating function f on G with the additional property that the sub graph of G induced by the set of all vertices of positive weight under f has no isolated vertices.
The minimum weight of a total Roman dominating function is called as the total Roman domination number of G and it is denoted by . A total Roman dominating function with minimum weight is called – function. If then G is called a total Roman graph.

TOTAL ROMAN DOMINATING FUNCTION
The Roman dominating function of a graph was defined by E. J. Cockayne et.al. [5]. The definition of a Roman dominating function was motivated by an article in Scientific American by Ian Stewart [8] entitled Defend The Roman Empire! and suggested even earlier by C.
S. ReVelle [3]. Domination number and Roman domination number in an interval graph with consecutive cliques of size 3 are studied by C. Jaya Subba Reddy, M. Reddappa and B. Maheswari [2].

INTERVAL GRAPH
Let be an interval
family, where each is an interval on the real line and = ] for Here is called the left end point and is called the right end point of . Without loss of generality, we assume that all end points of the intervals in are distinct numbers between 1 and 2n. Two intervals i = ] and j = [ ] are said to
intersect each other if either or . The
intervals are labelled in the increasing order of their right end points.
Let be a graph. G is called an interval graph if there is a 11 correspondence between and such that two vertices of are joined by an edge in if and only if their corresponding intervals in intersect. If is an interval in the corresponding vertex in is denoted by .
Consider the following interval family.
I3 I6
I1
I4
I2 I5
The corresponding interval graph is
v1
v2
v6
v3
, are total
dominating sets of respectively. Further we can show that all these sets are minimum total dominating sets. Therefore the total domination numbers of are for
and for
If k =2, then
For and for
, and for
and for
are minimum total dominating sets of . So the total domination numbers are for and
for respectively.
Similarly for k =3 we have
Then the minimum total dominating sets are
for
;
for
;
for
;
for
.
Hence for and for
v5
Thus for
v4 for
for
In what follows we consider interval graphs of this type. That is the interval graph which has consecutive cliques of size 3. We denote this type of interval graph by . The total domination and total Roman domination number is studied in the following for the interval graph .

RESULTS
Theorem 4.1: Let be the Interval graph with n vertices and no isolated vertices, where n . Then the total domination number of is
for n
for
for
for .
Hence we get that the general form of total dominating sets of
as
for for
where respectively.
Proof: Let be the Interval graph with vertex set
and no isolated
for
for
for for
vertices, where n .
Suppose k =1. Then We can easily see that is a total dominating set of .
Now for we see that
and for and for
for for
and so on.
Thus for n
g(v) g(v) g(v) g(v)
for
vV '
vV0
vV1
vV2
where respectively.
Theorem 4.2: Let be the interval graph with n vertices and no isolated vertices, where . Then
Proof: Let be the interval graph with n vertices and no isolated vertices, where . Let
{ } be the vertices of .
Then it is clear that is the total dominating set when
n = 3, 4 and is the total dominating set when
and is the total dominating set for
.
That is
Theorem 4.3: Let be the interval graph with n vertices and no isolated vertices, where n . Then the total Roman domination number of is
for
= for
.
where respectively.
Proof: Let be the interval graph with n vertices and no isolated vertices, where n .
Since is a minimum total dominating set of , we have
. Further , since . This implies that
.Thus
is the minimum weight of , where is a total Roman dominating function.
Therefore
Case 2: Suppose , where .
Now we proceed as in Case 1.
Let ,
;
.
Clearly is a minimum total dominating set of . Here we observe that the set dominates .
Therefore becomes a total Roman dominating function of .
Now
Therefore f (v) f (v) f (v) f (v) .
Let the vertex set of
vV vV
vV vV
be .
Case 1: Suppose , where .
Let and let be the ordered partition of induced by f where
for Then there exist a 1 correspondence between the functions
and the ordered pairs of
.Thus we write .
Let ;
;
0 1 2
= = 4 +2.
Let be a total Roman
dominating function of . Then we can show as in Case 1, that is a minimum weight of for the total Roman dominating function .
Thus
Case 3: Suppose , where . Now we proceed as in Case 1.
Let ;
=V{ }
. ;
By Theorem 4.1, we see that is a minimum total dominating set of . Further the set dominates . In addition the induced sub graph on is a sub graph of
with no isolated vertices.
Therefore is a total Roman dominating function of .
Now
Therefore f (v) f (v) f (v) f (v) .
.
Obviously is a minimum total dominating set of . Here we observe that the set dominates .
Therefore becomes a total Roman dominating function of .
Now
vV vV0
vV1
vV2
Therefore f (v) f (v) f (v) f (v) .
= = 4 +2
vV vV0
vV1
vV2
Let be a total Roman dominating function of , where dominates . Then
= = 4 +4.
Let be a total Roman
dominating function of . Then similar lines to Case 1, we
can show that is a minimum weight of for the total Roman dominating function .
Thus
Case 4: Suppose , where . Now we proceed as in Case 1.
Let ;
;
.
Again is a minimum total dominating set of . Further the
.
Here is a minimum total dominating set of and we observe that the set dominates .
Therefore becomes a total Roman dominating function of .
Now
Therefore f (v) f (v) f (v) f (v) .
set dominates .
vV vV0
vV1
vV2
Therefore becomes a total Roman dominating function of .
Now
Therefore f (v) f (v) f (v) f (v) .
= = 4 +4.
Let be a total Roman
dominating function of . Then we can show as in Case 1, that is a minimum weight of for the total Roman dominating function .
vV vV0
vV1
vV2
Thus
= = 4 +4.
Let be a total Roman
dominating function of . In similar lines to Case 1, we can show that is a minimum weight of for the total Roman dominating function .
Thus
Case 5: Suppose ,where . Now we proceed as in Case 1.
Let ;
;
.
Clearly is a minimum total dominating set of . Here we
Case 7: Suppose , where .
Now we proceed as in Case 1.
Let ,
.
We have seen that is a minimum total dominating set of and we observe that the set dominates .
Therefore becomes a total Roman dominating function of .
Now
Therefore f (v) f (v) f (v) f (v) .
observe that the set dominates .
vV vV0
vV1
vV2
Therefore becomes a total Roman dominating function of .
Now
Therefore f (v) f (v) f (v) f (v) .
= = 4 +4.
Let be a total Roman
dominating function of . In similar lines to Case 1, we can show that is a minimum weight of for the total Roman dominating function .
vV vV0
vV1
vV2
Thus
= = 4 +4.
Let be a total Roman
dominating function of . Then we can show as in Case 1, that is a minimum weight of for the total Roman dominating function .
Thus
Case 6: Suppose , where . Now we proceed as in Case 1.
Let ;
Theorem 4.4: Let be the interval graph with n vertices and no isolated vertices, where . Then the total Roman domination number is
for
= 4 for
Proof: Let be the interval graph with n vertices and no isolated vertices, where . Let
{ } be the vertices of .
Case 1: Suppose . Let be the vertices of
.
Let ;
; .
Obviously is a minimum total dominating set of and the set dominates .
Therefore is a total Roman dominating function of .
Therefore f (v) f (v) f (v) f (v) .
Proof : Let be the interval graph with n vertices and no isolated vertices, where .
Then it is clear that when we have seen by Theorem 4.4 and Theorem 4.2 that
Thus
Theorem 4.6: Let be the interval graph with n = 7 vertices and no isolated vertices. If . Then
is a total Roman graph.
vV vV0
vV1
vV2
Proof: Let be the interval graph with n = 7 vertices and
= 0 + 1 2×1 = 3.
Suppose n = 7.
no isolated vertices.
Thus .
Case 2: Suppose . Let be the vertices of .
Let ; ;
.
Here is a minimum total dominating set of and the set dominates . In similar lines to Case 1, we
get .
Case 3: Suppose . Let be the
vertices of .
Let ; ;
.
Again is a minimum total dominating set of and the set dominates . In similar lines to Case
1, we get .
Case 4: Suppose Let , be the
vertices of .
Let ; ;
.
Here is a minimum total dominating set of and the set
dominates .
Therefore is a total Roman dominating function of .
Therefore f (v) f (v) f (v) f (v) .
Then we have and . Thus .
Hence is a total Roman graph.
Theorem 4.7: Let be the interval graph with n vertices and no isolated vertices, where n . Then is a total Roman graph, for ,
, where
respectively.
Proof: Let be the interval graph with n vertices and no isolated vertices, where n .
Case 1: Suppose
respectively.
Then by Theorem 4.3, we have the total Roman domination number as
= 2 .
Thus is a total Roman graph.
Case 2: Suppose
, where
respectively.
Then by Theorem 4.3, we have the total Roman domination number as
= 2 .
Therefore is a total Roman graph.
vV vV0
Thus
vV1
= 0 2×2 = 4
vV2
Theorem 4.8: Let be the interval graph with n vertices and no isolated vertices, where n . Then is a total Roman graph if and only if there exist a function
Case 5: Suppose Let be
the vertices of .
Let ; ;
.
Again is a minimum total dominating set of and the set dominates . In similar lines to Case 4, we get
.
Theorem 4.5: Let be the interval graph with n vertices and no isolated vertices, where . Then
with .
Proof: Let be the interval graph with n vertices and no isolated vertices, where n . Suppose is a total Roman graph. Let be a function of .
Then we know that dominates and
dominates V. In addition the induced sub graph is a sub graph of with no isolated vertices.
Hence =
= . But is a total Roman graph. So
= 2 . Then it follows that , which
establishes Theorem 4.3. v9
Conversely, suppose there is a function 2
of such that . By the v8
definition of function, we have dominates V
and since , it follows that dominates V. In addition the induced sub graph is a sub graph of
with no isolated vertices. As is a minimum total v7
dominating set, we have = . By the definition of 2
function we have = = 0
= 2 . v6
v10
0
0 v1
0
v2
Hence is a total Roman graph, which also establishes Theorem 4.3.

ILLUSTRATIONS
Illustration 1: n = 6
0
v5 2
v3 2
v4 0
I3 I6
I1 I4
I2 I5
f (v)
vV
and
; ; = V{ }
Therefore .
V1
0 v2
V6 0
0
v3
2
v5
2 v4 0
and
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f (v)
vV
; ; =V{ }
.
Appl. Anal. Discrete math. vol.10, 2016, pp.501517.

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Therefore .
Illustration 2: n = 10
I3 I6

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I1 I4
I2 I5
I7 I10
I8
graphs: Advanced Topics, Marcel Dekkar, Inc., New York, 1998.

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