Characterization of Trees with Equal Total Edge Domination and Double Edge Domination Numbers
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Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.5, 2012
Characterization of Trees with Equal Total Edge Domination and
Double Edge Domination Numbers
M.H.Muddebihal A.R.Sedamkar*
Department of Mathematics, Gulbarga University, Gulbarga-585106, Karnataka, INDIA
E-mail of the corresponding author: mhmuddebihal@yahoo.co.in
Abstract
A total edge dominating set of a graph G is a set D of edges of G such that the sub graph D has no isolated
edges. The total edge domination number of G denoted by γ t (G ) , is the minimum cardinality of a total edge
'
dominating set of G . Further, the set D is said to be double edge dominating set of graph G . If every edge of G is
dominated by at least two edges of D . The double edge domination number of G , denoted by, γ d (G ) ,
'
is the
minimum cardinality of a double edge dominating set of G . In this paper, we provide a constructive
characterization of trees with equal total edge domination and double edge domination numbers.
Key words: Trees, Total edge domination number, Double edge domination number.
1. Introduction:
In this paper, we follow the notations of [2]. All the graphs considered here are simple, finite, non-trivial,
undirected and connected graph. As usual p = V and q = E denote the number of vertices and edges of a
graph G , respectively.
In general, we use X to denote the sub graph induced by the set of vertices X and N ( v ) and
N [ v ] denote the open and closed neighborhoods of a vertex v , respectively.
The degree of an edge e = uv of G is defined by deg e = deg u + deg v − 2 , is the number of edges
adjacent to it. An edge e of degree one is called end edge and neighbor is called support edge of G .
A strong support edge is adjacent to at least two end edges. A star is a tree with exactly one vertex of degree
greater than one. A double star is a tree with exactly one support edge.
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For an edge e is a rooted tree T , let C ( e ) and S ( e ) denote the set of childrens and descendants of e
respectively. Further we define S [e] = S (e) U {e} . The maximal sub tree at e is the sub tree of T induced
by S [e] , and is denoted by Te .
A set D ⊆ E is said to be total edge dominating set of G , if the sub graph D has no isolated edges. The
total edge domination number of G , denoted by γ t
'
(G ) , is the minimum cardinality of a total edge dominating set
of G . Total edge domination in graphs was introduced by S.Arumugam and S.Velammal [1].
A set S ⊆ V is said to be double dominating set of G , if every vertex of G is dominated by at least two
vertices of S . The double domination number of G , denoted by γ d (G ) , is the minimum cardinality of a double
dominating set of G . Double domination is a graph was introduced by F. Harary and T. W. Haynes [3]. The concept
of domination parameters is now well studied in graph theory (see [4] and [5]).
Analogously, a set D ⊆ E is said to be double edge dominating set of G , if every edge of G is dominated
by at least two edges of D . The double edge domination number of G , denoted by γ d
'
(G ) , is the minimum
cardinality of a double edge dominating set of G .
In this paper, we provide a constructive characterization of trees with equal total edge domination and
double edge domination numbers.
2. Results:
Initially we obtain the following Observations which are straight forward.
Observation 2.1: Every support edge of a graph G is in every γ t' (G ) set.
Observation 2.2: Suppose every non end edge is adjacent to exactly two end edge, then every end edge of a graph
G is in every γ d (G ) set.
'
Observation 2.3: Suppose the support edges of a graph G are at distance at least three in G , then every support
edge of a graph G is in every γ d (G ) .
'
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3. Main Results:
Theorem 3.1: For any tree T , γ d (T ) ≥ γ t' (T ) .
'
Proof: Let q be the number of edges in tree T . We proceed by induction on q . If diam (T ) ≤ 3 . Then T is either
a star or a double star and γ d
'
(T ) = 2 = γ t' (T ) . Now assume that diam (T ) ≥ 4 and the Theorem is true for
every tree T ' with q ' < q . First assume that some support edge of T , say ex is strong. Let ey and ez be the end
edges adjacent to ex and T ' = T − e . Let D ' be any γ d (T ' ) - set. Clearly ex ∈ D ' , where D ' is a total edge
'
dominating set of tree T . Therefore γ t' (T ' ) ≤ γ t' (T ) . Now let S be any γ d (T )
'
- set. By observations 2 and 3, we
have e y , e x , e z ∈ S . Clearly, S − {ey } is a double edge dominating set of tree T ' . Therefore
γ d (T ' ) ≤ γ d (T ) − 1.
' '
Clearly, γ d (T ) ≥ γ d (T ' ) + 1 ≥ γ t' (T ' ) + 1 ≥ γ t' (T ) + 1 ≥ γ t' (T ) ,
' '
a contradiction.
Therefore every support edge of T is weak.
Let T be a rooted tree at vertex r which is incident with edge er of the diam (T ) . Let et be the end
edge at maximum distance from er , e be parent of et , eu be the parent of e and ew be the parent of eu in the
rooted tree. Let Tex denotes the sub tree induced by an edge ex and its descendents in the rooted tree.
Assume that degT (eu ) ≥ 3 and eu is adjacent to an end edge ex . Let T ' = T − e and D ' be the
γ t' (T ' ) - set. By Observation 1, we have eu ∈ D ' . Clearly, D ' U {e} is a total edge dominating set of tree T . Thus
γ t' (T ) ≤ γ t' (T ' ) + 1 . Now let S be any γ d (T ) -
'
set. By Observations 2 and 3, et , e x , e, eu ∈ S . Clearly,
S − {e, et } is a double edge dominating set of tree T ' . Therefore γ d (T ') ≤ γ d (T ) − 2 .
' '
It follows that
γ d (T ) ≥ γ d (T ' ) + 2 ≥ γ t' (T ' ) + 2 ≥ γ t' (T ) +1 ≥ γ t' (T ) .
' '
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Now assume that among the decedents of eu there is a support edge say ex , which is different from e . Let
T ' = T − e and D ' be the γ t' (T ' ) - set containing no end edges. Since ex must have a neighbor in D ' , thus
eu ∈ D ' . Clearly D ' U {e} is a total edge dominating set of tree T and hence γ t' (T ) ≤ γ t' (T ' ) + 1 . Now let S be
any γ d (T ) - set. By Observations 2 and 3, we have et , e, e x ∈ S . If eu ∈ S , then S − {e, et } is the double edge
'
dominating set of tree T . Further assume that eu ∉ S . Then S U {eu } − {e, et } is a double edge dominating set of
'
tree T . Therefore γ d (T ' ) ≤ γ d (T ) − 1 .
' '
Clearly, it follows that, γ d (T ) ≥ γ d (T ' ) + 1 ≥ γ t' (T ') + 1 ≥ γ t' (T ) .
' '
Therefore, we obtain γ d (T ) ≥ γ t' (T ) .
'
To obtain the characterization, we introduce six types of operations that we use to construct trees with equal total
edge domination and double edge domination numbers.
Type 1: Attach a path P to two vertices u and w which are incident with eu and ew respectively of T where
1
eu , ew located at the component of T − ex ey such that either ex is in γ d set of T or ey is in γ d - set of T .
' '
Type 2: Attach a path P2 to a vertex v incident with e of tree T , where e is an edge such that T − e has a
component P .
3
Type 3: Attach k ≥ 1 number of paths P3 to the vertex v which is incident with an edge e of T where e is an
edge such that either T − e has a component P2 or T − e has two components P2 and P4 , and one end of P4 is
adjacent to e is T .
Type 4: Attach a path P3 to a vertex v which is incident with e of tree T by joining its support vertex to v , such
that e is not contained is any γ t' - set of T .
Type 5: Attach a path P4 ( n ) , n ≥ 1 to a vertex v which is incident with an edge e , where e is in a γ d - set of T
'
if n = 1 .
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Type 6: Attach a path P5 to a vertex v incident with e of tree T by joining one of its support to v such that
T − e has a component H ∈{P3 , P4 , P6 }.
Now we define the following families of trees
Let ℑ be the family of trees with equal total edge domination number and double edge domination number. That is
ℑ= {T / T is a tree satisfying γ t' (T ) = γ d ( T )} .
'
We also define one more family as
ℜ = {T / T is obtained from P3 by a finite sequence of type - i operations where
1 ≤ i ≤ 5} .
We prove the following Lemmas to provide our main characterization.
Lemma 3.2: If T ∈ ℑ and T is obtained from T by Type-1 operation, then T ∈ ℑ .
' '
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ' ) . By Theorem 1, γ t' (T ) ≤ γ d (T ) , we only need to prove that
' '
γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching the path P to two vertices u and w which are
1
incident with eu and ew as eu and ew respectively. Then there is a path ex ey in T ' such that either ex is in γ d -
' ' '
set of T and T ' − ex ey has a component P5 = eu eew ex or ey is in γ d - set of T ' and T ' − ex ey has a
'
component P6 = eu eew ex ex . Clearly,
'
γ t' (T ' ) = γ t' (T ) − 1 .
Suppose T ' − ex ey contains a path P5 = eu eew ex then S ' be the γ d - set of T ' containing ex . From Observation 2
'
and by the definition of γd
'
- set, we have S ' I {eu , e , ew , ex } = {eu , ew } or {eu , e} . Therefore
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S = ( S ' − {eu , e , ew }) U {eu , e, ew } is a double edge dominating set of T with S = S ' + 1 = γ d (T ') + 1 .
' ' '
Clearly, γ t' (T ) = γ t' (T ' ) + 1 = γ d (T ' ) + 1 = S > γ d (T ) .
' '
Now, if T ' − ex ey contains a path P6 = eu e ew ex ex . Then S ' be the γ d - set of T ' containing ey . By Observation 2
' '
and by definition of γd
'
- set, we have S I { eu , e, ew , ex , ex } = { eu , ew , ex } .
' ' '
Therefore
S = ( S ' − {eu , ew }) U {eu , e, ew } is a double edge dominating set of T with S = S ' + 1 = γ d (T ' ) + 1 .
' ' '
Clearly, γ t' (T ) = γ t' (T ') + 1 = γ d (T ' ) +1 = S ≥ γ d (T ) .
' '
Lemma 3.3: If T ' ∈ ℑ and T is obtained from T ' by Type-2 operation, then T ∈ ℑ .
Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t' (T ) ≤ γ d (T ) ,
' ' ' ' ' '
we only need to prove that
γ t' (T ) ≥ γ d (T ) . Assume that T
' '
is obtained from T by attaching a path P2 to a vertex v which is incident with
e of T ' where T ' − e has a component P3 = ewex . We can easily show that γ t' (T ) = γ t' (T ' ) + 1 . Now by
definition of γd
'
- set, there exists a γd
'
- set, D ' of T ' containing the edge e . Then D ' U {eu } forms a double
'
edge dominating set of T . Therefore γ t' (T ) = γ t' (T ' ) + 1 = γ d (T ' ) + 1 = D ' U {eu } ≥ γ d (T ) .
' ' '
Lemma 3.4: If T ∈ ℑ and T is obtained from T by Type - 3 operation, then T ∈ ℑ .
' '
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' '
that γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching m ≥ 1 number of paths P3 to a vertex v
which is incident with an edge e of T ' such that T ' − e has a component P3 or two components P2 and P4 . By
definition of γ t' - set and γd -
'
set, we can easily show that γ t
'
(T ) ≥ γ t' (T ' ) + 2m and γ d (T ') + 2m ≥ γ d (T ) .
' '
Since γ t (T ' ) = γ d (T ') , it follows that γ t' (T ) ≥ γ t' (T ') + 2m = γ d (T ') + 2m ≥ γ d (T ) .
' ' ' '
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Lemma 3.5: If T ' ∈ ℑ and T is obtained from T ' by Type - 4 operation, then T ∈ ℑ .
Proof: Since T ∈ ℑ , we have γ t (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' ' ' '
that γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching path P3 to a vertex v incident with e in
T ' such that e is not contained in any γ t' -set of T ' and T '− e has a component P4 . For any γ d - set, S ' of T ' ,
'
S ' U {ex , ez } is a double edge dominating set of T . Hence γ d (T ' ) + 2 ≥ γ d (T ) . Let D be any γ t' - set of
' '
T containing the edge eu , which implies ey ∈ D and D I {e, ex , ez } =1 .
If e ∉ D , then D I E (T ' ) = D − 2 = γ t' (T ) − 2 ≥ γ t' (T ' ) , since D I E (T ' ) is a total edge
T '. since γ t (T ' ) = γ d (T ') , that γ t (T ) ≥ γ t (T ') + 2
' ' ' '
dominating set of Further it follows
= γ d (T ') + 2 ≥ γ d (T ) .
' '
If e∈D , then D I {e, ex , ez } = {e} and D I E (T ' ) = D −1 = γ t' (T ) −1 ≥ γ t' (T ') , since
D I E (T ') is a total edge dominating set of T ' . Suppose γ t' (T ) ≤ γ d (T ) − 1 , then by γ d (T ' ) = γ t' (T ') , it
' '
follows that γ d (T ) ≥ γ t' (T ) + 1≥ γ t' (T ') + 2 = γ d (T ') + 2 ≥ γ d (T ') .
' ' '
Clearly,
D I E (T ') = γ t' (T ) − 1 = γ t' (T ') and D I E (T ') is a total edge dominating set of T '
containing e .
'
Therefore, it gives a contradiction to the fact that e is not in any total edge dominating set of T and
hence γ t (T ) ≥ γ d (T ) .
' '
Lemma 3.6: If T ∈ ℑ and T is obtained from T by Type - 5 operation, then T ∈ ℑ .
' '
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' '
that γ t' (T ) ≥ γ d (T ) . Assume that T
'
is obtained from T ' by attaching path P4 ( n ) , n ≥ 1 to a vertex v incident
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with e in T ' such that e is in γ d - set for n = 1 . Clearly, γ t' (T ) ≥ γ t' (T ' ) + 2 n . If n ≥ 2 , then by
'
γ t' (T ' ) = γ d (T ') , it is obvious that γ t' (T ) ≥ γ t' (T ') + 2n = γ d (T ') + 2 n ≥ γ d (T )
' ' '
. If n = 1 , then D ' be a
γd
'
- set of T ' containing e . Thus D ' U {ez , e x } is a double edge dominating set of T . Hence
γ t' (T ) ≥ γ t' (T ') + 2 = γ d (T ') +2 = S ' U {ez , ex } ≥ γ d (T ).
' '
Lemma 3.7: If T ∈ ℑ and T is obtained from T ' by Type - 6 operation, then T ∈ ℑ .
'
Proof: Since T
'
∈ ℑ , we have γ t' (T ' ) = γ d (T ') . By Theorem 1, γ t' (T ) ≤ γ d (T ) , hence we only need to prove
' '
that γ t' (T ) ≥ γ d (T ) .
'
Assume that T is obtained from T ' by attaching path a path P5 to a vertex v which is
incident with e . Then there exists a subset D of E (T ) as γ t' - set of T such that D I NT ' (e) ≠ φ for n = 1 .
Therefore D I E (T ') is a total edge dominating set of T ' and D I E (T ') ≥ γ t' (T ') . By the definition of double
edge dominating set, we have γ d (T ') + 3 ≥ γ d (T )
' '
. Clearly, it follows that
γ t' (T ) = D = D I E ( P6 ) + D I E (T ') > 3 + γ t' (T ') = 3 + γ d (T ') ≥ γ d (T ) .
' '
Now we define one more family as
Let T be the rooted tree. For any edge e ∈ E (T ) , let C ( e ) and F ( e ) denote the set of children edges and
descendent edges of e respectively. Now we define
C ' (e) = {eu ∈ C (e) every edge of F [ eu ] has a distance at most two from e in T } .
Further partition C ' (e) into C1' (e) , C2' (e) and C3' (e) such that every edge of Ci ' (e) has edge
degree i in T , i = 1, 2 and 3 .
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Lemma 3.8: Let T be a rooted tree satisfying γ t' (T ) = γ d (T )
'
and ew ∈ E (T ) . We have the following
conditions:
If C (ew ) ≠ φ , then C1 (ew ) = C3 (ew ) = φ .
' ' '
1.
If C3 (ew ) ≠ φ , then C1' (ew ) = C2' (ew ) = φ and C3' (ew ) = 1 .
'
2.
3. If C (ew ) = C
'
(ew ) ≠ C1' (ew ) , then C1' (ew ) = C3' (ew ) = φ .
Proof: Let C1' (ew ) = {ex1 , ex2 ,...........exl } , C2' (ew ) = {ey1 , ey2 ,...........eym } and C3' (ew ) =
{ez1 , ez2 ,...........ezn } such that C1' (ew ) = l , C2' (ew ) = m and C3' (ew ) = n . For every i = 1, 2,..., n , let
eu be the end edge adjacent to e z in T and T ' = T − {ex1 , ex2 ,..., exl , eu1 , eu2 , ..., eun } .
i i
For (1): We prove that if m ≥ 1 , then l + n = 0 . Assume l + n ≥ 1 . Since m ≥ 1 , we can have a γd
'
- set S of T
such that ew ∈ S and a γ t' - set D of
'
T ' such that ew ∈D . Clearly S − {ex1 , ex2 ,..., exl , eu1 , eu2 , ..., eun } is a
'
double edge dominating set of T' and D ' is a total edge dominating set of T . Hence
γ t' (T ' ) = D ' ≥ γ t' (T ) = γ d (T ) = S > S − {ex , ex ,..., ex , eu , eu , …, eu } ≥ γ d (T ' ) , it gives a contradiction
'
1 2
'
l 1 2 n
with Theorem 1.
For (2) and (3): Either if C3' (ew ) ≠ φ or if C (ew ) = C ' (ew ) ≠ C1' (ew ) . Then for both cases, m + n ≥ 1 . Now
T ' such that ew ∈ D . Then D ' is also a total edge dominating set of T . Hence
'
select a γ t' - set D of
'
γ t' (T ' ) = D ' ≥ γ t' (T ) . Further by definition of γd
'
- set and by Observation 2, there exists a γd
'
-set S of T
which satisfies S I {ey1 , ey2 ,..., eym , ez1 , ez2 ,..., ezn } = φ . Then ( S I E (T ' )) U {ew } is a γ d -set of T ' . Hence
'
γ d (T ' ) ≤ ( S I E (T ' )) U {ew } ≤ S − (l + n) + 1 = γ d (T ) − (l + n) + 1 = γ t' (T ) − (l + n) + 1 .
' '
If n ≥ 1 , then γ d (T ') ≤ γ t' (T ) ≤ γ t' (T ' ) ≤ γ d (T ' ) , the last inequality is by Theorem 1. It follows that l + n = 1
' '
and ew ∉ S . Therefore l = 0 and n = 1 . From Condition 1, we have m = 0 . Hence 2 follows.
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If C (ew ) = C (ew ) ≠ C1' (ew ) , then m + n ≥ 1 . By conditions 1 and 2, l = 0 . Now we show that n = 0 .
'
Otherwise, similar to the proof of 2, we have ew ∉ S , n = 1 and m = 0 . Since C (ew ) = C ' (ew )
and deg (ew ) = 2 , for double edge domination, ew, ez ∈S , a contradiction to the selection of S .
Lemma 3.9: If T ∈ ℑ with at least three edges, then T∈ℜ.
Proof: Let q= E(T) . Since T ∈ ℑ , we have γ t
'
(T ) = γ d (T ) . If diam (T ) ≤ 3 , then T is either a star or a
'
double star and γ t
'
(T ) = 2 = γ d (T ) . Therefore T ∈ ℜ . If diam (T ) ≥ 4 , assume that the result is true for all
'
T ' with E(T ) = q < q .
' '
trees
We prove the following Claim to prove above Lemma.
Claim 3.9.1: If there is an edge ea ∈E(T) such that T − ea contains at least two components P3 , then T ∈ ℜ .
= eb eb and ec ec are two components of T − ea . If T =T −{eb , eb}, then use D ' and S to
' ' ' '
Proof: Assume that P3
denote the γ t' -set of T ' containing ea and γ d - set of T , respectively. Since ea ∈ D ' , D ' U {eb } is a total edge
'
dominating set of T and hence γ t (T
' '
) ≥ γ t' (T) −1. Further since S is a γd
'
- set of T , S I {ea , eb , eb } = {ea , eb }
' '
by the definition of γd
'
- set. Clearly, S I E (T
'
) is a double edge dominating set of T ' and hence γ t' (T ' ) ≥
γ t' (T) −1=γ d (T) −1= S I E(T ') ≥ γ d (T ') . By using Theorem 1, we get γ t' (T ' ) = γ d' (T ')
' '
and so T '∈ ℑ .
By induction on T ' , T ' ∈ℜ . Now, since T is obtained from T ' by type - 2 operation, T ∈ ℜ .
By above claim, we only need to consider the case that, for the edge ea , T − ea has exactly one component P3 . Let
P = eu e ew ex ey ez ...er be a longest path in T having root at vertex r which is incident with er .
Clearly, C(ew) =C ' (ew) ≠ C1' (ew ) .By 3 of Lemma 6, C1' (ew ) = C3' (ew ) = φ . Hence P4 = eu e ew is a
component of T − ex . Let n be the number of components of P4 of S ( ex ) in T such that an end edge of every
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P4 is adjacent to ex . Suppose S ( ex ) in T has a component P4 with its support edge is adjacent to ex . Then it
consists of j number of P3 and k number of P2 components. By Lemma 6, m, j ∈ {0,1} and k ∈ {0,1, 2} .
Denoting the n components P4 of the sub graph S ( ex ) in T with one of its end edges is adjacent to an edge
ex in T by P4 = eui ei ewi , 1 ≤ i ≤ n . We prove that result according to the values of {m, j , k} .
Case 1: Suppose m = j = k =0. Then S ( ex ) = P4 ( n ) , n ≥ 1 in T. Further assume that
T ' = T − S [ ex ] , then 2 ≤ E (T '
) < q . Clearly, γ t
'
(T ' ) ≥ γ t' (T ) − 2n . Let S be a γ d - set of T such that S
'
contains as minimum number of edges of the sub graph S ( ex ) as possible. Then ex ∉ S and S I S[ex ] = 2n
by the definition of γd
'
- set. Clearly S I E (T ') is a double edge dominating set of T '
and
hence γ t (T ' ) ≥ γ t' (T ) − 2n = γ d (T ) −2n = S I E (T ' ) ≥ γ d (T ) . By Theorem 1, γ t' (T ' ) = γ d (T ' ) and
' ' ' '
S I E (T ' ) is a double edge dominating set of T ' . Hence T ' ∈ ℑ . By applying the inductive hypothesis to T ' ,
T ' ∈ℜ .
If n ≥ 2 , then it is obvious that T is obtained from T ' by type - 5 operation and hence T ∈ ℜ .
If n = 1 . Then S ( ex ) = P4 = eu e ew in T which is incident with x of an edge ex and
S I{eu , e, ew, ex} = {eu , ew} . To double edge dominate, ex , ey ∈ S and so ey ∈ S I E (T ' ) , which implies
that ey is in some γ d - set of T ' . Hence T is obtained from T ' by type-5 operation and T ∈ℜ .
'
Case 2: Suppose m ≠ 0 and by the proof of Lemma 6, m = 1 and j = k = 0 . Denote the component P4 of
S ( ex ) in T whose support edge is adjacent to ex in T by P4 = ea eb ec and if T ' = T − {ea , eb , ec } . Then,
clearly 3 ≤ E (T '
) ≤ q . Let S be a γ '
d - set of T which does not contain eb .
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Now we claim that ex is not in any γ t' - set of T ' . Suppose that T ' has a γ t' - set containing ex which is denoted
'
by D , then D ' U {eb } is a total edge dominating set of T . Clearly, γ t' (T ' ) ≥ γ t' (T ) −1 . Since eb ∉ S then
S I E (T ' ) is a double edge dominating set of T '. Hence
γ t' (T ' ) ≥ γ t' (T ) −1 = γ d (T ) −1 = S I E (T ' ) + 1≥ γ d (T ' ) + 1 ,
' '
which gives a contradiction to the fact that
γ t' (T ' ) ≤ γ d (T ' ) . This holds the claim and therefore T
'
can be obtained from T ' by type-4 operation.
Now we prove that T ' ∈ℜ . Let D ' be any γ t' - set of T ' . By above claim, ex ∉ D ' . Since D ' U {ex , eb } is a total
edge dominating set of T , γ t
'
(T ) ≥ γ (T ) − 2 . Further since e
'
t
'
b ∉ S , S I E (T ' ) is a double edge dominating
set of T ,
'
γ t' (T ' ) ≥ γ t' (T ) − 2 = γ d (T ) − 2 = S I E (T ' ) ≥ γ d (T ' ) .
' '
Therefore by Theorem 1, we get
γ t' (T ' ) = γ d (T ' ) ,
'
which implies that T ' ∈ ℑ . Applying the inductive hypothesis on T ' , T '∈ℜ and hence
T ∈ℜ .
Case 3: Suppose j ≠0 and by the proof of Lemma 6, m = k = 0 . Let i {
T ' =T − Un=1 eui ,ei ,ewi } . Clearly,
3 ≤ E (T ' ) < q and T is obtained from T ' by type - 3 operation.
We only need to prove that T '∈ℜ . Suppose D ' ⊂ E (T ' ) be a γ t' - set of T ' . Then
( { })
D' U Un=1 ei ,ewi
i is a total edge dominating set of T and hence. γ t' (T ' ) ≥ γ t' (T ) − 2n . Since T − ex has a
component P = ea eb , we can choose
3
S⊆ E(T) as a γd
'
- set of T containing ex . Then S I E(T ' ) is a γ d' -
set of T '
and hence γ d (T ) = S = 2n +
'
S I E (T ' ) ≥ 2n + γ d (T ' ) . Clearly, it follows that,
'
γ t' (T ' ) ≥ γ d (T ' ) . Therefore, by Theorem 1, we get, γ t' (T ' ) = γ d (T ' ) and hence T ' ∈ ℑ . Applying the inductive
' '
hypothesis on T ' , we get T '∈ℜ .
83
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Vol.2, No.5, 2012
Case 4: Suppose k ≠ 0 . Then by Lemma 6, k ∈{1,2} and so m = j = 0 . We claim k = 1 . If not, then k = 2 . We
denote the two components P2 of S ( ex ) by ex and ex in T . Let T ' = T − ex . Clearly, γ t' (T ' ) = γ t' (T ) . Let S
' '' ''
be a γd
'
- set of T containing {e w1 }
, ew2 ,..., ewn . By Observation 2, {ex , ex } ⊆ S . Since S I E(T ) is a double
' '' '
edge dominating set of T' with S I E(T ' ) = γ d (T) −1,
'
we have
γ t' (T ' ) = γ t' (T ) = γ d (T ) > γ d (T ) −1 ≥ γ d (T ' ) , which is a contradiction to the fact that γ t' (T ' ) ≤ γ d (T ' ) .
' ' ' '
Sub case 4.1: For n ≥ 2 . Suppose T
'
i {
=T − Un=1 eui ,ei ,ewi } . Then T is obtained from T '
by type - 3 operation.
Now by definition of γ t' - set and γd -
'
set, it is easy to observe that γ t (T
' '
) + 2 (n − 1) = γ t' (T ) and
γ d (T ' ) + 2(n − 1) = γ d (T )
' '
. Hence γ t' (T ' ) = γ d (T ' ) and T ' ∈ ℑ .
'
Applying the inductive hypothesis on T ',
T ' ∈ℜ and hence T ∈ ℜ .
Sub case 4.2: For n = 1 . Denote the component P2 of S ( ex ) by ex in T . Suppose S ( e y ) − S [ ex ] has a
'
component H∈{P , P , P } in T , then T ' = T − S [ ex ] . Therefore we can easily check that T
3 4 6 is obtained from
T ' by type-6 operation. Now by definition of γd -
'
set, γ d (T ' ) + 3 = γ d (T ) .
' '
For any γ t' '
- set D of T ',
D ' U {e , ew , ex } is a total edge dominating set of T . Clearly, γ t' (T ' ) ≥ γ t' (T ) − 3 = γ d (T ) − 3 = γ d (T ' ) . By
' '
Theorem 1, we get γ t' (T ' ) = γ d (T ' ) and T ' ∈ ℑ .
'
Applying inductive hypothesis on T , T ∈ ℜ and hence
' '
T ∈ℜ .
Now if the sub graph S ( ey ) − S [ ex ] has no components P3 , P4 or P6 . Then we consider the structure
S ( ey ) in T . By above discussion, S ( ey ) consists of a component P6 = eu e ew ex ex and g number of
'
of
components of P2 , denoted by {e , e ,..., e } . Assume l = 2 . Then, let T
1 2 g
'
= T − S ey . It can be easily checked
that γ t' (T ' ) + 4 ≥ γ d (T ) = γ d (T ' ) + 5 ,
' '
which is a contradiction to the fact that γ t' (T ' ) ≤ γ d (T ' ) .
'
Hence
g ≤ 1.
84
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Suppose T ' = T − {eu , ex } . Here we can easily check that γ t' (T ' ) + 1 = γ t' (T ) . Let S
'
be a γ d - set of
'
T such that S contains as minimum edges of S ey as possible and S I S [ex ] = {eu , ew ex } . Then
'
S ' = ( S − {eu , ew ex }) U {e, ex }
'
is a double edge dominating set of T '.
Therefore γ t
'
(T ' ) = γ t' (T ) −1 = γ d (T ) −1 = S ' ≥ γ d (T ' ) , which implies that γ t' (T ' ) = γ d (T ' ) where S ' is a
' ' '
double edge dominating set of T ' . Hence T ' ∈ ℑ . Applying inductive hypothesis to T ' , T ' ∈ ℜ .
If g = 0 , then degT ( ey ) = 2 . Since ex ∉S , to double edge dominate ey , ey ∈S . Therefore
ey is in the double edge dominating set D ' of T ' . Hence T is obtained from T ' by type-1 operation. Thus
T ∈ℜ .
If g = 1 , then degT ( ey ) = 3 . Since ex ∉S to double edge dominate ey , we have ey ∉S and
ez ∈S , by the selection of S . Therefore ez is in the double edge dominating set S ' of T ' . Hence T is obtained
from T by type-1operation. Thus T ∈ ℜ .
'
By above all the Lemmas, finally we are now in a position to give the following main characterization.
Theorem 3.10: ℑ = ℜ U { P3}
References
S. Arumugan and S. Velammal, (1997), “Total edge domination in graphs”, Ph.D Thesis, Manonmaniam Sundarnar
University, Tirunelveli, India.
F. Harary, (1969), “Graph theory”, Adison-wesley, Reading mass.
F. Harary and T. W. Haynes, (2000), “Double domination in graph”, Ars combinatorics, 55, pp. 201-213.
T.W. Haynes, S. T. Hedetniemi and P. J. Slater, (1998), “Fundamentals of domination in graph”, Marcel Dekker,
New York.
T. W. Haynes, S. T. Hedetniemi and P. J. Slater, (1998), “Domination in graph, Advanced topics”, Marcel Dekker,
New York.
Biography
A. Dr.M.H.Muddebihal born at: Babaleshwar taluk of Bijapur District in 19-02-1959.
85
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.5, 2012
Completed Ph.D. education in the field of Graph Theory from Karnataka. Currently working as a Professor in
Gulbarga University, Gulbarga-585 106 bearing research experience of 25 years. I had published over 50 research
papers in national and international Journals / Conferences and I am authoring two text books of GRAPH THEORY.
B. A.R.Sedamkar born at: Gulbarga district of Karnataka state in 16-09-1984.
Completed PG education in Karnataka. Currently working as a Lecturer in Government Polytechnic Lingasugur,
Raichur Dist. from 04 years. I had published 10 papers in International Journals in the field of Graph Theory.
86
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