IRJET-Entire Domination in Jump Graphs

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 05 Issue: 06 | June-2018 www.irjet.net p-ISSN: 2395-0072
© 2018, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1749
ENTIRE DOMINATION IN JUMP GRAPHS
N.pratap babu Rao1, Sweta. N2
Department of Mathematics S.G. degree college KOPPAL(Karnataka)INDIA
Department of Mathematics Veerasaiva college Ballari (Karnataka) INDIA
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ABSTRACT:
The vertices and edges of a graph J(G) are called the element of J(G). A set X of elements in J(G) is an entire dominating
set if every element not inX ix either adjacent or incident to at least one element in X The entire domination number (J(G)) is
the order of a smallest entire dominating set in J(G) In this paper exact values of (J(G)) for some standard graphs are
obtained Also, bounds on )) and Nordhaus- Gaddam type results are established.
INTRODUCTION;
The graph considered here are finite, connected, undirected without loops or multiple edges. We denote by √(J(G))
and )) the vertex set and the edge set of J(G) respectively. For any undefined term or notation in this paper see
Harary[3]. The study of dominating sets in graph was begun by Ore[7] and Berge[5]. The entire domination number was
defined by Kulli[4].
The open neighborhood N(v) ( N(e)) of a vertex ( an edge e) is the set of vertices (edges) adjacent to v(e). The closed
neighborhood N[v] 9 N[e]) of a vertex 9an edge e0 is N(v) {v} ( N(e) {e} ). The open entire neighborhood n(x) of an edge x
is the set of elements either adjacent or incident to x. the closed entire neighborhood n[x] of an element x is n(x) {x}. (J(G))
denoes the maximum degree of J(G). The degree of an edge e=uv is defined as deg u +deg v -2.The maximum edge degree of
J(G) is denoted by ’(J(G)), we will employ the following notation ┌ x ┐ ( └ x ┘ ) to denote the smallest (largedst) integer
greater(lesser) than equal tox
A set D of vertices in J(G) is a dominating set if every vertex not in D is adjacent to atleast one vertexin V(J(G)) – D. The
domination number √(J(G)) is the order of a smallest dominating set in J(G).
A set F of edges of J(G) is an edge dominating set if every edge not in F is adjacent to at least one edge in E(J(G)) – F.
The edge domination number √’(J(G)) of J(G) is the smallest edge dominating set in J(G)..
We now obtained a relation between the domination, edge domination and entire domination number of a graph.
Theorem 1; For any graph J(G) ( √(J(G)) + √’(J(G))) /2 ≤ (J(G)) ≤ √(J(G)) + √’(J(G)).
Further the upper bound attains if there exists a minimum entire dominating set X= D f satisfying.
i) N[D} = V(J(G)), N[F] = E(J(G)) and N[v] = N[e] =
ii) Deg v = (J(G)), deg e = ’ (J(G)) for all in D and e in F.
Proof; First we establish the lower bound . Let X =D F be a minimum entire dominating set of J(G). for each edge e=uv in F
Choose a vertex u or v, not both and F’ be the collection of such vertices Clearly D F’ is a dominating set ,
There fore
√(J(G)) ≤ | E F’ |
= | D F|
= (J(G))……(1)
Now for each vertex u in D choose exactly one edge e incident with u and let D’ be the collection of such edges. Clearly D’ F
is an edge dominating set. Therefore

√’(J(G)) ≤ | D’ F |
= |D F |
= (J(G)) ……..(2)
From (1) and (2) follows
√(J(G)) + √’(J(G)) ≤ 2 (J(G)).
Therefore
√(J(G)) + √’(J(G)) /2 = (J(G))
Now for the upper bound, let D and f be the minimum dominating and edge dominating sets respectively.
Then D F is an entire dominating set. Thus
(J(G)) ≤ | D F |
= √(J(G)) + √ ‘(J(G)).
Theorem 2; For any connected jump graph J(G).
P – q ≤ (J(G)) ≤ p - ┌
)
┐
For the lower bound is attained if and only if J(G) is a star.
Proof; First we establish the upper bound. Let v be a vertex of degree (J(G)).Let F be the set of independent edges in <N(v)>.
Then V(J(G)) F – N(v) is an entire dominating set. Also |F| ≤ └
)
┘ Therefore
(J(G)) ≤ | V(J(G)) F – N(v) |
≤ p + └
)
┘ - (J(G))
≤ p - ┌
)
┐
Now for the lower bound, let X be a minimum entire dominating set of J(G). Then
P + q - |X| = | V(J(G)) E(J(G)) – X|
≤ | V(J(G)) E(J(G)) | - 1
≤ p + q-(p-q)
≤ 2q.
Then (J(G)) ≥ p-q.
Suppose (J(G)) = p-q Then p-q ≥1 and from the above inequalities it follows that p-q=1 This shows that J(G) is a star.
Conversely, suppose J(G) is a star obliviously (J(G)) = p-q.
Theorem 3; For any jump graph J(G)
(J(G)) ≥
)
) )

i)
ii) For any elemrnt x in (V E)_X there is anelement y in X such that
n(x) X ={y}
iii) |n(x)| = 2, (J(G)) for every x in X
Proof; This follows from Theorem A and the notation of totalgraph if there exists a minimum entire dominating set satisfying
(i) (ii) and (iii) the bound is attained.
Theorem 4; For any connected J(G) of order p
(J(G)) ≤ ┌ ┐
Proof; We prove the result by induction on p if p≤ 4 then the result can be verified. Assume the result is true for all connected
graphs J(G) and p-2 vertices. Let J(G) be a connected graph then p vertices. Let u and v denote either two adjacent vertices or
two non adjacent vertices having a common neighbor w such that J(G)= J(G’) – {u v} is connected. Let X be the minimum
entire dominating set of J(G). Then either X {w] or X {u v} is an entire dominating set of J(G’). Then,
(J(G’)) ≤ |X| + 1
≤ ┌ ┐+ 1
= ┌ ┐
Finally we establish Nordhaus-Gaddum type results.
Theorem 5 ; For any connected graph J(G) with p vertices
(J(G)) + (J( ̅)) ≤ ┌ ┐
(J(G)) + (J( ̅)) ≤ p ┌ ┐
Proof; J(G) is complete, then J ( ̅ ) is totally disconnected (J( ̅)) = p
There fore
(J(G)) + (J( ̅)) = ┌ ┐ + p
= ┌ ┐
And (J(G)) . (J( ̅)) = p ┌ ┐
Theorem6; Let J(G) and J( ̅ ) be connected complete graph then,
(J(G)) + (J( ̅)) ≤ p + 1
(J(G)) . (J( ̅)) ≤ (p+1)2 / 4

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[3] F. Harary, Graph Theory Addison Wesley reading Mass (1969)
[4] V.R. Kulli, On entire domination number, Second Conf.Ramanujan Math.Soc., Madras (1987).
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computing Utilitas Mathematica, Winnipeg (1997) 489-509.
[6] E.A. Nordhaus and J.W Gaddum, On complementary graphs Amer.Math.Monthly 63 (1956) 175-177.
[7] O.Ore Theory of Graphs.Amer.Math.Soc., Colloq.pul., 38 Providence (1962)
[8] H>B>Walikar, B.D.Acharya and E.Sampthkumar, Recent DDevelopments in the Theory of Domination in Graphs.MRI
Lecture Notes in Math.1 (1979).
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IRJET-Entire Domination in Jump Graphs

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