IRJET- Total Domatic Number of a Jump Graph
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This document discusses total domination and total domatic numbers in jump graphs. It begins with definitions of key graph theory terms like domination sets, total domination number, and total domatic number. It then presents several theorems, including ones that prove if G is a graph of order p > 4, then the total domatic number of G plus the total domatic number of its complement is less than or equal to p - 2. The document characterizes the class of all regular jump graphs for which the total domatic number of G plus the total domatic number of the complement equals p - 2. It shows these graphs must be isomorphic to cycles of certain lengths, disjoint unions of cycles, or two specific
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 05 Issue: 11 | Nov 2018 www.irjet.net p-ISSN: 2395-0072
© 2018, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 571
TOTAL DOMATIC NUMBER OF A JUMP GRAPH
N. Pratap Babu Rao
Department of Mathematics S.G. Degree College Koppal (Karnataka) INDIA
------------------------------------------------------------------------***-----------------------------------------------------------------------
ABSTRACT - Let dt and ̅t denotes the total domatic
number of a jump graph J(G) and its complement J( ̅). In
this paper wecharactrize the class of all regular jump
graphs for which dt + ̅t =p-2 where p is the order of J(G).
Key words: Total domination, connected domination,
Total domatic number, connected domatic number
Mathematics subject-Classification:05C
1. Introduction
By a graph we mean a finite undirected graph without
loops or multiple edges. Terms not defined here are used
in Harary [1]
Let J(G)+(v,E) be a jump graph of order p and size q. a
subset S of V(J(G)) is dominating set of G if every vertex in
V(J(G))-S is adjacent to some vertex in S. Let J(G) be a jump
graph without isolated vertices. A subset S of V(J(G)) is
called a total dominating set of J(G) is called the total
domination number of J(G) and is denoted by √t(J(G)). The
maximum order of a partition of V(J(G)) into total
dominating sets of J(G) is called the total domatic number
of J(G) and is denoted by dt(J(G)).
Let J(G) be a connected jump graph. a dominating set S of
J(G) is called a connected dominating , if <S> is connected.
The cardinality of a minimum connected dominating set is
called the connected domination number of J(G) and is
denoted by √c(J(G))..The minimum order of a partition of
V into connected dominating set is called the domatic
number of J(G) and is denoted by dc. The total connected
domatic number of complement J( ̅) is denoted by ̅
t
(̅
c).The total(connected) domatic number of J( ̅). is
denoted by ̅t ( ̅c). The maximum and minimum degrees
of a vertex in jump graph J(G) are denoted by Δ and δ
respectively. For any real number x, └ x ┘ denotes the
largest integer less than or equal to x.
Cockayne, Dawes and Hedetniem i[2] have proved the
following .
Theorem1.1 [2] if g has p vertices no isolates and Δ < p-1
then v ≤ p-1 then dt + ̅t ≤ p-1 with inequality if and only
if G or ̅ is isomorphic to C4.
Hence it follows that if G is a graph of order p > 4 then dt +
̅t ≤ p - 2 we give an independent proof of this inequality
which enables us to obtain a characterization of all regular
graphs for which dt + ̅t = p-2. The characterization of non-
regular graph for which dt + ̅t = p - 2 will be repeated in a
subsequent paper.
We need the following theorems
Theorem 1.2 [2] For any graph without isolate dt ≤ δ
Theorem 1.3 [2] Let G be a regular graph of order p such
that both G and ̅ are connected. Then
dc + ̅c =p-2if and only if G or ̅ is isomorphic to C6 or G1
or G2 Where G1 and G2 are given in Figure 1
G1
G2
Figure 1
2. Main Results
Theorem 2.1 Let J(G) be any jump graph with at least 5
vertices no isolates and Δ ≤p-2 then dt + ̅t ≤ p-2
Proof; Since √t ≥2 dt ≤ └ p/2┘. If G disconnected jump
graph with n components, n≥2, we have √t ≥ 2n so that dt
≤└ p/2n┘ and dt + ̅t ≤ └ p/2 ┘ ≤ p – 2.Hence. we may
assume that both G and ̅ are connected.
If dt ≤ └ p/2 ┘ - 1 and ̅t ≤ └ p/2 ┘ - 1 then the result is
trivial. Hence we may assume without loss of generality
that dt = └ p/2 ┘. We consider the following cases
Case i) p is even and J(G) is r-regular. Then J ( ̅) is ̅ -
regular. Where ̅ = p – 1 – r. Since dt ≤ δ we have r ≥
p/2 so that ̅ ≤ (p/2) – 1. Hence ̅t ≥ 3 and ̅t ≤ └ P/3
┘
Thus dt + ̅t ≤ └p/2┘ + └ p/3 ┘ ≤ p-2 provided p >6
when p=6 r=3 and ̅ = 2 so that ̅ is isomorphic to G
hence dt + ̅t = 3 + 1 = 4 = p – 2.](https://image.slidesharecdn.com/irjet-v5i11111-181201084102/85/IRJET-Total-Domatic-Number-of-a-Jump-Graph-1-320.jpg)
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 05 Issue: 11 | Nov 2018 www.irjet.net p-ISSN: 2395-0072
© 2018, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 572
Case ii) p is even and J(G) is non regular Then δ ≥ p/2, Δ ≥
(p/2) + 1 so that ̅ ≥ (p/2) – 2 Hence ̅t ≤ ( p/2 ) – 2 and
dt + ̅t ≤ p-2
Case iii): p is odd Then Δ ≥ └ ┘ + 1 and henc it follows
that ̅t ≥ 3 Then ̅t ≤ └ P/3 ┘
So that dt + ̅t ≤ └p/2┘ + └ p/3 ┘ ≤ p-2 we now
proceed to characterize the class of all regular graphs for
which dt + ̅t = p-2.
Theorem 2.2: Let J(G) be disconnect4ed r-regular jump
graph with at least 5 vertices.
Then dt + ̅t = p-2. If and only if J(G) is isomorphic to
2C3, 2C4, 3K2 or 2K
proof; Suppose J(G) has n components. Then it follows
from Theorem2.1 that dt + ̅t = p-2. If and only if └ p/2n
┘ + └ p/2 ┘ = p – 2 and hence n ≤ 3 when n=2 , p=6 or 8
when p=6 J(G) is isomorphic to 2C3 and p=8 J(G) is
isomorphic to 2K2
.
Converse is trivial.
Theorem 2.3 Let J(G) be connected 2-regular jump
graph with atleast 5 vertices
Then dt + ̅t + p-2. If and only if J(G) is isomorphic to
C6 or C8
Proof; Trivial
Theorem 2.4 Let J(G) be a r-regular jump graph such
that J(G) and its complement
J( ̅ ) are c onnected and r, ̅ ≥ 3 Then dt + ̅t + p-2. If
J(G) and J( ̅ ) is isomorphic if
G1 or G2 where G1 and G2 are the jump graphs given in
Figure 1.
Proof; Suppose dt + ̅t = p-2 it follows from theorem
2.1 that
└p/2┘ + └ p/3 ┘ = p-2 hence 5 ≤ p ≤ 12
p 6,11 Since r, ̅ ≥ 3 we have p≥ 8 so that p= 8,9,10 or
12. Now we claim that
dt or ̅t = └ p/2 ┘, otherwise we have dt = ̅t = └ p/2 ┘. –
1 and hence p is even so that dt ≤ r and ̅t ≤ ̅.Hence
either r or ̅ is (p/2)-1 suppose ̅ =(p/2)-1. Then ̅t ≤ └
p/3 ┘ and hence dt + ̅t < p-2, which is a contradiction.
Thus dt b or ̅t = └ p/2 ┘ suppose dt = └ p/2 ┘. If p=9 it
follows that r=5 and ̅ = 3which is impossible. Hence
p=8,10, 12 suppose p=10 V= Ui=1 Vi and | Vi| = 2 and |Vi |
is total dominating set in J(G). Then r ≥ 5 and ̅ ≤ 4 and
̅̅̅ ≥ 3 we claim that ̅t ≥ 4 suppose { v1, v2 , } is a total
dominating set in J( ̅ ) and let v2v1, v2v3 ̅ ))
we assume that vi I = 1, 2, 3, let w2 be the other
vertex of V2 Since v1 v2 , v2v3 ̅ )) it follows that
v1w1 ̅ )) Similarly
v2w2 2 ̅ )) and w2 is not dominated by any
vertex of S in ̅̅̅ ≥ 4 and ̅̅̅ ≤ 2 so that dt + ̅t <
p-2 which is a contradiction. Then p 10 by a similar
argument p ≠ 12 Then p = 8 since dt = 4 it follows that dc
= 4. Also r-4 and and ̅ =3, ̅̅̅=2. Hence by Theporem 1.3
J(G) is isomorphic to J(G1) or J(G2).
REFERECES
[1] F. Harary Graph Theory Addison Wesley Reading Mass
(1972)
[2] E.J Cockayne, R.M Dawes and S.T. Hedetniemi,
Networks 10 (1980)21-219
[3] J. Paulraj Joseph and S. Armugam J. Ramanujan
math.Sco,9 (19940 No.1 69-77.
[4] S.Armugam and A. T Thuraiswamy, Indian.J.pure appl.
Math.29(5) (1998) 513-515](https://image.slidesharecdn.com/irjet-v5i11111-181201084102/85/IRJET-Total-Domatic-Number-of-a-Jump-Graph-2-320.jpg)
IRJET- Total Domatic Number of a Jump Graph
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 05 Issue: 11 | Nov 2018 www.irjet.net p-ISSN: 2395-0072 © 2018, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 571 TOTAL DOMATIC NUMBER OF A JUMP GRAPH N. Pratap Babu Rao Department of Mathematics S.G. Degree College Koppal (Karnataka) INDIA ------------------------------------------------------------------------***----------------------------------------------------------------------- ABSTRACT - Let dt and ̅t denotes the total domatic number of a jump graph J(G) and its complement J( ̅). In this paper wecharactrize the class of all regular jump graphs for which dt + ̅t =p-2 where p is the order of J(G). Key words: Total domination, connected domination, Total domatic number, connected domatic number Mathematics subject-Classification:05C 1. Introduction By a graph we mean a finite undirected graph without loops or multiple edges. Terms not defined here are used in Harary [1] Let J(G)+(v,E) be a jump graph of order p and size q. a subset S of V(J(G)) is dominating set of G if every vertex in V(J(G))-S is adjacent to some vertex in S. Let J(G) be a jump graph without isolated vertices. A subset S of V(J(G)) is called a total dominating set of J(G) is called the total domination number of J(G) and is denoted by √t(J(G)). The maximum order of a partition of V(J(G)) into total dominating sets of J(G) is called the total domatic number of J(G) and is denoted by dt(J(G)). Let J(G) be a connected jump graph. a dominating set S of J(G) is called a connected dominating , if <S> is connected. The cardinality of a minimum connected dominating set is called the connected domination number of J(G) and is denoted by √c(J(G))..The minimum order of a partition of V into connected dominating set is called the domatic number of J(G) and is denoted by dc. The total connected domatic number of complement J( ̅) is denoted by ̅ t (̅ c).The total(connected) domatic number of J( ̅). is denoted by ̅t ( ̅c). The maximum and minimum degrees of a vertex in jump graph J(G) are denoted by Δ and δ respectively. For any real number x, └ x ┘ denotes the largest integer less than or equal to x. Cockayne, Dawes and Hedetniem i[2] have proved the following . Theorem1.1 [2] if g has p vertices no isolates and Δ < p-1 then v ≤ p-1 then dt + ̅t ≤ p-1 with inequality if and only if G or ̅ is isomorphic to C4. Hence it follows that if G is a graph of order p > 4 then dt + ̅t ≤ p - 2 we give an independent proof of this inequality which enables us to obtain a characterization of all regular graphs for which dt + ̅t = p-2. The characterization of non- regular graph for which dt + ̅t = p - 2 will be repeated in a subsequent paper. We need the following theorems Theorem 1.2 [2] For any graph without isolate dt ≤ δ Theorem 1.3 [2] Let G be a regular graph of order p such that both G and ̅ are connected. Then dc + ̅c =p-2if and only if G or ̅ is isomorphic to C6 or G1 or G2 Where G1 and G2 are given in Figure 1 G1 G2 Figure 1 2. Main Results Theorem 2.1 Let J(G) be any jump graph with at least 5 vertices no isolates and Δ ≤p-2 then dt + ̅t ≤ p-2 Proof; Since √t ≥2 dt ≤ └ p/2┘. If G disconnected jump graph with n components, n≥2, we have √t ≥ 2n so that dt ≤└ p/2n┘ and dt + ̅t ≤ └ p/2 ┘ ≤ p – 2.Hence. we may assume that both G and ̅ are connected. If dt ≤ └ p/2 ┘ - 1 and ̅t ≤ └ p/2 ┘ - 1 then the result is trivial. Hence we may assume without loss of generality that dt = └ p/2 ┘. We consider the following cases Case i) p is even and J(G) is r-regular. Then J ( ̅) is ̅ - regular. Where ̅ = p – 1 – r. Since dt ≤ δ we have r ≥ p/2 so that ̅ ≤ (p/2) – 1. Hence ̅t ≥ 3 and ̅t ≤ └ P/3 ┘ Thus dt + ̅t ≤ └p/2┘ + └ p/3 ┘ ≤ p-2 provided p >6 when p=6 r=3 and ̅ = 2 so that ̅ is isomorphic to G hence dt + ̅t = 3 + 1 = 4 = p – 2.
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 05 Issue: 11 | Nov 2018 www.irjet.net p-ISSN: 2395-0072 © 2018, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 572 Case ii) p is even and J(G) is non regular Then δ ≥ p/2, Δ ≥ (p/2) + 1 so that ̅ ≥ (p/2) – 2 Hence ̅t ≤ ( p/2 ) – 2 and dt + ̅t ≤ p-2 Case iii): p is odd Then Δ ≥ └ ┘ + 1 and henc it follows that ̅t ≥ 3 Then ̅t ≤ └ P/3 ┘ So that dt + ̅t ≤ └p/2┘ + └ p/3 ┘ ≤ p-2 we now proceed to characterize the class of all regular graphs for which dt + ̅t = p-2. Theorem 2.2: Let J(G) be disconnect4ed r-regular jump graph with at least 5 vertices. Then dt + ̅t = p-2. If and only if J(G) is isomorphic to 2C3, 2C4, 3K2 or 2K proof; Suppose J(G) has n components. Then it follows from Theorem2.1 that dt + ̅t = p-2. If and only if └ p/2n ┘ + └ p/2 ┘ = p – 2 and hence n ≤ 3 when n=2 , p=6 or 8 when p=6 J(G) is isomorphic to 2C3 and p=8 J(G) is isomorphic to 2K2 . Converse is trivial. Theorem 2.3 Let J(G) be connected 2-regular jump graph with atleast 5 vertices Then dt + ̅t + p-2. If and only if J(G) is isomorphic to C6 or C8 Proof; Trivial Theorem 2.4 Let J(G) be a r-regular jump graph such that J(G) and its complement J( ̅ ) are c onnected and r, ̅ ≥ 3 Then dt + ̅t + p-2. If J(G) and J( ̅ ) is isomorphic if G1 or G2 where G1 and G2 are the jump graphs given in Figure 1. Proof; Suppose dt + ̅t = p-2 it follows from theorem 2.1 that └p/2┘ + └ p/3 ┘ = p-2 hence 5 ≤ p ≤ 12 p 6,11 Since r, ̅ ≥ 3 we have p≥ 8 so that p= 8,9,10 or 12. Now we claim that dt or ̅t = └ p/2 ┘, otherwise we have dt = ̅t = └ p/2 ┘. – 1 and hence p is even so that dt ≤ r and ̅t ≤ ̅.Hence either r or ̅ is (p/2)-1 suppose ̅ =(p/2)-1. Then ̅t ≤ └ p/3 ┘ and hence dt + ̅t < p-2, which is a contradiction. Thus dt b or ̅t = └ p/2 ┘ suppose dt = └ p/2 ┘. If p=9 it follows that r=5 and ̅ = 3which is impossible. Hence p=8,10, 12 suppose p=10 V= Ui=1 Vi and | Vi| = 2 and |Vi | is total dominating set in J(G). Then r ≥ 5 and ̅ ≤ 4 and ̅̅̅ ≥ 3 we claim that ̅t ≥ 4 suppose { v1, v2 , } is a total dominating set in J( ̅ ) and let v2v1, v2v3 ̅ )) we assume that vi I = 1, 2, 3, let w2 be the other vertex of V2 Since v1 v2 , v2v3 ̅ )) it follows that v1w1 ̅ )) Similarly v2w2 2 ̅ )) and w2 is not dominated by any vertex of S in ̅̅̅ ≥ 4 and ̅̅̅ ≤ 2 so that dt + ̅t < p-2 which is a contradiction. Then p 10 by a similar argument p ≠ 12 Then p = 8 since dt = 4 it follows that dc = 4. Also r-4 and and ̅ =3, ̅̅̅=2. Hence by Theporem 1.3 J(G) is isomorphic to J(G1) or J(G2). REFERECES [1] F. Harary Graph Theory Addison Wesley Reading Mass (1972) [2] E.J Cockayne, R.M Dawes and S.T. Hedetniemi, Networks 10 (1980)21-219 [3] J. Paulraj Joseph and S. Armugam J. Ramanujan math.Sco,9 (19940 No.1 69-77. [4] S.Armugam and A. T Thuraiswamy, Indian.J.pure appl. Math.29(5) (1998) 513-515