J. Comp. & Math. Sci. Vol.2 (4), 617-620 (2011)
The Maximal Equitable Domination of a Graph P. N. VINAY KUMAR1 and D. SONER NANDAPPA2 1
Faculty of Mathematics, Government Science College, Hassan-573 201, Karnataka, India 2 Department of Mathematics, University of Mysore, Mysore-570 0 06, Karnataka, India ABSTRACT
Let G = (V , E ) is a simple graph with vertex set V ( G ) and edge set
E (G ) .
A subset D of V is said to be an equitable
dominating set of G , if every vertex u ∈ D equitably dominates every vertex v ∈V − D . An equitable dominating set D of a graph G is maximal if, V − D is not an equitable dominating set e of G . The maximal equitable domination number γ m ( G ) of G is the minimum cardinality of a maximal equitable dominating set. In this paper, we introduce and discuss the concept of maximal equitable domination number of G . We obtain some bounds on
γ me ( G ) and also, we investigate the relationship of γ me ( G ) with other graph parameters. Keywords: Equitable dominating set, equitable domination number, maximal equitable dominating set, maximal equitable domination number. Mathematics Subject Classification (2000): 05C.
3.1 INTRODUCTION Suppose that the graph G = (V , E ) is an undirected finite graph without loops or multiple edges. A set D of vertices in a graph G is a dominating set of G if every vertex V − D is adjacent to some vertex in D . The set D is said to be equitable dominating set of G if every vertex
v ∈V − D is adjacent to some vertex u ∈ D and The maximal d (u ) − d ( v ) ≤ 1 . domination number concept was introduced by V. R. Kulli and B. Janakiram6. A dominating set D of a graph G is maximal if V − D is not a dominating set of G . The maximal domination number γ m ( G ) of G is the minimum cardinality of a maximal
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P. N. Vinay Kumar, et al., J. Comp. & Math. Sci. Vol.2 (4), 617-620 (2011)
dominating set. For any undefined terms we refer to F. Harary2. In this paper, we introduce and study the concept maximal e equitable domination number γ m ( G ) of a graph G . An equitable dominating set D of a graph G is a maximal equitable dominating set if, V − D is not an equitable The maximal dominating set of G . e equitable domination number γ m ( G ) of G is the minimum cardinality of a maximal equitable dominating set. Clearly, e . γ m (G ) ≤ γ m (G ) 3.2 RESULTS Theorem 3.2.1: Let G be any graph with e p vertices then γ e ( G ) ≤ γ m (G ) . Proof: We know that γ ( G ) ≤ γ e ( G ) and also, every maximal equitable dominating set is an equitable dominating set of G . This proves the result. Theorem 3.2.2: For any graph G , 2, if m - n ≤ 1 m + n,if m - n ≥ 2
e γm ( K m, n ) =
I.
II.
III.
n 3 +1, if n ≡1,2( mod3) e γ m ( Cn ) = n + 2, if n ≡ 0( mod3) 3 n
e γm ( Pn ) = + 1 3
IV.
n 3 + 2, if n ≡ 1,2( mod3) e . γm (Wn ) = n + 3, if n ≡ 0( mod3) 3
Theorem 3.2.31: Let G be a graph without equitable isolated points and D is a minimal equitable dominating set of G then V − D is an equitable dominating set of G . Theorem 3.2.4: A minimal equitable dominating set D of a graph G is a maximal equitable dominating set if and only if G contains equitable isolated vertices. Proof: Suppose that G contains an equitable isolated vertex v . Then, v is in every equitable dominating set D of G and hence V − D is not an equitable dominating set of G . This proves that D is maximal. Conversely, if D is a minimal equitable dominating set then by the above theorem 3.2.3, G contains an equitable isolated vertices. Theorem 3.2.5: Let G be any graph on p vertices and without equitable isolated vertices then γ me ( G ) = p if and only if G ≅ K p with p ≥ 3 . Suppose that γ me ( G ) = p and G ≠ K p . Then, G has at least three vertices
Proof:
G is u , v and w , since p ≥ 3 . As complete, if suppose u and v are adjacent then at least one of them is not adjacent with w . Suppose that u is not adjacent with w . Then,
V − {u} is
a
maximal
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equitable
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P. N. Vinay Kumar, et al., J. Comp. & Math. Sci. Vol.2 (4), 617-620 (2011)
dominating set of G , which contradiction. Hence, G ≅ K p .
is
Conversely, suppose that G ≅ K p .
Then,
a
each of the vertices equitably dominates all other vertices. Hence, maximal equitable dominating set of G contains all p vertices. That is, γ me ( G ) = p .
e 2 ≤ γm (G) ≤ p .
Theorem 3.2.7: For any graph G with equitable isolated vertices, γ me ( G ) = p if and only if G ≅ K p with I. p ≥ 3. II.
(G ) = m + n
G = K m,n
if and with m − n ≥ 2 .
only
if
Case i): Suppose that G ≅ K p and D be a minimal equitable dominating set of G . Then by theorem3.2.4, D is maximal. This implies that D = p . Hence, γ me ( G ) = p . Hence the result. ii):
Suppose
that
G = K m , n with
{u1, u2 ,L, um }
m − n ≥ 2.
Let
{v1, v2 ,L, vn }
be two disjoint partition of
vertex
set
of
( )
d ( ui ) − d v j ≥ 2
( )
d ( ui ) − d v j ≥ 2 .
e Hence, γ m (G ) = m + n .
G.
As with
e vertices then γ m (G ) ≤ γ e (G ) + δ e (G ) .
Proof: Let D be a minimum equitable dominating set of G and v be a vertex of e minimum equitable degree δ ( G ) . Then, either v ∈ D or v ∉ D . If v ∈ D then,
D ∪ N e ( v ) is a maximal equitable dominating set of G and hence,
e γm (G ) ≤ D ∪ N e ( v ) = γ e (G ) + δ e (G ) .
On the other hand, if
v ∉ D , then
D ∪ N [ v] is a maximal equitable dominating set of G and hence e
Proof: To prove the theorem, we consider the following cases.
Case
and all v j ’s since,
Theorem 3.2.8: If G is any graph on p
Corollary 3.2.6: Let G be any connected ( p, q ) − graph with p ≥ 3 and without equitable isolated vertices. Then,
e γm
1 ≤ j ≤ n . If D * is a minimum equitable dominating set of G , then by theorem3.2.4, D * is maximal. Thus, D * contains all ui ’s
and
m − n ≥ 2,
1 ≤ i ≤ m and
γ me ( G ) ≤ γ e ( G ) + δ e ( G ) ≤ D ∪ N e [ v ] . Theorem 3.2.9: Let G be a graph without equitable isolated vertices and with pendant e vertices. Then, γ m ( G ) = γ m ( G ) .
Proof: Let D be a maximal equitable dominating set of G and v be a pendant vertex of G . Then, either v ∈ D or v ∉ D . If v ∈ D then, v is adjacent to some vertex in Theorem 3.2.10: Let G be a graph without equitable isolated vertices then,
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P. N. Vinay Kumar, et al., J. Comp. & Math. Sci. Vol.2 (4), 617-620 (2011)
γ me ( G ) ≤ αoe ( G ) + 1 . Proof: Let S be a maximal equitable independent set. Then, for any vertex v ∈ S ,
V − S ∪ {v} is a maximal equitable dominating set of G . This proves the result.
Theorem 3.2.11: Let G be any connected ( p, q ) − graph with p ≥ 3 then,
p e q ≤ γ m ( G ) ≤ 2q − p + 2 . Proof:
REFERENCES
Since for any connected
( p, q ) −
p graph G , ≤ 2 and by corollary 3.2.6, q p γ me ( G ) ≥ 2 . Hence, we have γ me ( G ) ≥ . q Since G is a connected p ≥ 3 , we have
( p, q ) −
vertex v ∈V (T ) with S ∪ {v} is a maximal equitable dominating set of T . This proves the result. Suppose that the bound is not attained then, S , the set of all cut vertices becomes maximal equitable independent set of T , a contradiction. Hence, each of the cut vertices is adjacent to an end vertex.
graph with
γ me ( G ) ≤ p = 2 ( p − 1) − p + 2 ≤ 2q − p + 2 . Theorem 3.2.12: Let T be a tree without equitable isolated vertices. Then,
γ me ( G ) ≤ m + 1 . Further, the bound is attained if and only if each cut vertex is adjacent to an end vertex of T .
Proof: Let S be the set of all cut vertices of T with S = m . Then, there exists an end
1. K. M. Dharmalingam, Studies in graph theory – equitable domination and bottleneck domination, Ph.D. Thesis (2006). 2. F. Harary, Graph theory, Addison Wesley, Reading Mass (1972). 3. T.W. Haynes, S.T. Hedetneimi and P.J. Slater, Fundamentals of domination in graphs, Marcel Dekker Inc., New York, (1998). 4. T.W. Haynes, S.T. Hedetneimi and P.J. Slater, Domination in graphs: advanced topics, Marcel Dekker Inc., New York, (1998). 5. S.T. Hedetniemi and R.C. Laskar, Bibliography on domination in graphs and some basic definitions of domination parameters, Discrete Math., 86, 257-277 (1990). 6. V. R. Kulli and B. Janakiram, The maximal domination number of a graph, Graph Theory Notes of New York, New York Academy of Sciences, XXXIII, 11-13 (1997).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)