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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012

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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 adjacent to it. An edge

e = uv of G is defined by deg e = deg u + deg v − 2 , is the number of edges

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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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 For an edge

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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 by S

e is the sub tree of T induced

[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.

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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 a star or a double star and γ d '

every tree

(T ) = 2 = γ t' (T ) . Now assume that

diam (T ) ≤ 3 . Then T is either

diam (T ) ≥ 4 and the Theorem is true for

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

γ t' (T ' ) ≤ γ t' (T ) . Now let S

dominating set of tree T . Therefore

Clearly,

Therefore every support edge of

- set. By observations 2 and 3, we

γ d' (T ) ≥ γ d' (T ' ) + 1 ≥ γ t' (T ' ) + 1 ≥ γ t' (T ) + 1 ≥ γ t' (T ) ,

a contradiction.

T is weak.

Let T be a rooted tree at vertex edge at maximum distance from er , rooted tree. Let

γ d' (T )

S − {ey } is a double edge dominating set of tree T ' . Therefore

have e y , e x , e z ∈ S . Clearly,

γ d' (T ' ) ≤ γ d' (T ) − 1.

be any

r which is incident with edge er of the diam (T ) . Let et be the end

e be parent of et , eu be the parent of e and ew be the parent of eu in the

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} γ t' (T ) ≤ γ t' (T ' ) + 1 .

Now let S be any

γ d' (T ) -

is a total edge dominating set of tree T . Thus

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 ' .

γ d' (T ) ≥ γ d' (T ' ) + 2 ≥ γ t' (T ' ) + 2 ≥ γ t' (T ) +1 ≥ γ t' (T ) .

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Therefore

γ d' (T ') ≤ γ d' (T ) − 2 .

It follows that


Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 Now assume that among the decedents of

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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 .

Therefore, we obtain

Clearly, it follows that,

γ d' (T ) ≥ γ d' (T ' ) + 1 ≥ γ t' (T ') + 1 ≥ γ t' (T ) .

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

P1 to two vertices u and w which are incident with eu and ew respectively of T where

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 P3 . 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 adjacent to

e is T .

Type 4: Attach a path that

P2 or T − e has two components P2 and P4 , and one end of P4 is

P3 to a vertex v which is incident with e of tree T by joining its support vertex to v , such

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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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 Type 6: Attach a path

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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 incident with

is obtained from

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, '

Suppose and

T ' by attaching the path P1 to two vertices u and w which are

γ t' (T ' ) = γ t' (T ) − 1 .

T ' − ex ey contains a path P5 = eu eew ex then S ' be the γ d' - set of T ' containing ex . From Observation 2

by the

definition

of

γ d'

-

set,

we

have

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S ' I {eu , e , ew , ex } = {eu , ew } or {eu , e} . Therefore


Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012

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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,

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

Now, if and

γ t' (T ) = γ t' (T ' ) + 1 = γ d' (T ' ) + 1 = S > γ d' (T ) .

by

definition

of

γ d'

-

set,

have S I { eu , e, ew , ex , ex } = { eu , ew , ex } . '

we

'

'

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 ) . Assume that T

'

'

'

'

γ t' (T ) ≤ γ d' (T ) ,

is obtained from T by attaching a path

we only need to prove that

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

γ d'

γ d'

- set, there exists a

edge dominating set of T . Therefore

- set,

D ' of T ' containing the edge e . Then D ' U {eu' } forms a double

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 that

'

'

∈ ℑ , we have γ t' (T ' ) = γ d' (T ') . By Theorem 1, γ t' (T ) ≤ γ d' (T ) , hence we only need to prove

γ t' (T ) ≥ γ d' (T ) . Assume that T

which is incident with an edge definition of Since γ t

'

γ t' -

set and

γ d' -

is obtained from

T ' by attaching m ≥ 1 number of paths P3 to a vertex v

e of T ' such that T ' − e has a component P3 or two components P2 and P4 . By set, we can easily show that γ t

'

(T ) ≥ γ t' (T ' ) + 2m and γ d' (T ') + 2m ≥ γ d' (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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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012

T ' ∈ ℑ and T is obtained from T ' by Type - 4 operation, then T ∈ ℑ .

Lemma 3.5: If

Proof: Since T ∈ ℑ , we have γ t

'

'

that

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(T ' ) = γ d' (T ') . By Theorem 1, γ t' (T ) ≤ γ d' (T ) , hence we only need to prove

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 dominating

set

of

D I E (T ' ) = D − 2 = γ t' (T ) − 2 ≥ γ t' (T ' ) , since D I E (T ' ) is a total edge T '.

Further

since γ t

'

(T ' ) = γ d' (T ') ,

it

follows

that γ t (T ) ≥ γ t '

'

(T ') + 2

= 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 ') .

D I E (T ') = γ t' (T ) − 1 = γ t' (T ') and D I E (T ') is a total edge dominating set of T Therefore, it gives a contradiction to the fact that hence γ t

'

Clearly, '

containing e .

e is not in any total edge dominating set of T

'

and

(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 that

'

'

∈ ℑ , we have γ t' (T ' ) = γ d' (T ') . By Theorem 1, γ t' (T ) ≤ γ d' (T ) , hence we only need to prove

γ t' (T ) ≥ γ d' (T ) . Assume that T

is obtained from

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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 with

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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 ) γ d'

. If n = 1 , then

D ' be a

T ' containing e . Thus D ' U {ez , e x } is a double edge dominating set of T . Hence

- set of

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 '

Proof: Since T that

'

T is obtained from T ' by Type - 6 operation, then T ∈ ℑ .

∈ ℑ , we have γ t' (T ' ) = γ d' (T ') . By Theorem 1, γ t' (T ) ≤ γ d' (T ) , hence we only need to prove

γ t' (T ) ≥ γ d' (T ) .

Assume that T is obtained from

incident with e . Then there exists a subset D of Therefore

T ' by attaching path a path P5 to a vertex v which is

E (T ) as

γ t' - set of

T such that D I NT ' (e) ≠ φ for n = 1 .

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 descendent edges of

e respectively. Now we define

C ' (e) = {eu ∈ C (e) Further partition degree i in T ,

C ( e ) and F ( e ) denote the set of children edges and

every edge of

F [ eu ] has a distance at most two from e in T } .

C ' (e) into C1' (e) , C2' (e) and C3' (e) such that every edge of Ci ' (e) has edge

i = 1, 2 and 3 .

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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 Lemma 3.8: Let T be a rooted tree satisfying

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γ t' (T ) = γ d' (T )

and

ew ∈ E (T ) . We have the

following

conditions:

Proof:

1.

If C (ew ) ≠ φ , then C1 (ew ) = C3 (ew ) = φ .

2.

If C3 (ew ) ≠ φ , then

3.

If C (ew ) = C

'

'

'

'

'

C1' (ew ) = C2' (ew ) = φ and C3' (ew ) = 1 .

(ew ) ≠ C1' (ew ) , then C1' (ew ) = C3' (ew ) = φ .

C1' (ew ) = {ex1 , ex2 ,...........exl } ,

Let

C2' (ew ) = {ey1 , ey2 ,...........eym }

C3' (ew ) =

and

{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 such that double

ew ∈ S edge

and a

γ t'

dominating

- set S of T

'

T ' such that ew ∈D . Clearly S − {ex1 , ex2 ,..., exl , eu1 , eu2 , ..., eun } is a

of

T'

- set D of set

γ d'

'

D ' is a total edge dominating set of

and

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 select a

γ t' -

'

set D of

γ t' (T ' ) = D ' ≥ γ t' (T ) which satisfies

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 '

. Further by definition of

γ d'

- set and by Observation 2, there exists a

γ d'

-set S of T

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 and

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

ew ∉ S . Therefore l = 0 and n = 1 . From Condition 1, we have m = 0 . Hence 2 follows.

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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 If C (ew ) = C

'

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(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 Proof: Let

q= E(T)

double star and γ t

'

trees

. Since T ∈ ℑ , we have γ t

'

T∈ℜ. (T ) = γ d' (T ) . If diam (T ) ≤ 3 , then T is either a star or a

(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 . '

'

We prove the following Claim to prove above Lemma. Claim 3.9.1: If there is an edge ea ∈E(T) such that Proof: Assume that P3 denote the

γ t' -set of

= eb eb' and ec ec' are two components of T − ea . If T =T −{eb , eb}, then use D ' and S to '

'

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 '

by the definition of

T − ea contains at least two components P3 , then T ∈ ℜ .

γ d'

'

) ≥ γ t' (T) −1. Further since S

- set. Clearly, S I E (T

'

is a

γ d'

- set of T ,

S I {ea , eb , eb' } = {ea , eb' }

) 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 ') By induction on

and so

T '∈ ℑ .

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

component of T

3 of Lemma 6,

C1' (ew ) = C3' (ew ) = φ . Hence P4 = eu e ew is a

− ex . Let n be the number of components of P4 of S ( ex ) in T such that an end edge of every 81


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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.

T ' = T − S [ ex ] , then 2 ≤ E (T

'

Then

S ( ex ) = P4 ( n ) , n ≥ 1

) < q . Clearly, γ

' t

hence γ t

'

γ d'

- set. Clearly

(T ' ) ≥ γ t' (T ) − 2n = γ d' (T ) −2n =

Further

assume

that

(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 by the definition of

T.

in

S ( ex ) as possible. Then ex ∉ S and S I S[ex ] = 2n

S I E (T ') is a double edge dominating set of T

'

and

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

S ( ex ) = P4 = eu e ew in T

n = 1 . Then

which is incident with

S I{eu , e, ew, ex} = {eu , ew} . To double edge dominate, ex , ey ∈ S that

and so

x of an edge ex and

ey ∈ S I E (T ' ) , which implies

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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ex is not in any γ t' - set of T ' . Suppose that T ' has a γ t' - set containing ex which is denoted

Now we claim that

D ' U {eb } is a total edge dominating set of T . Clearly, γ t' (T ' ) ≥ γ t' (T ) −1 . Since eb ∉ S then

'

by D , then

S I E (T ' ) is

a

double

edge

dominating

γ t' (T ' ) ≥ γ t' (T ) −1 = γ d' (T ) −1 = S I E (T ' ) + 1≥ γ d' (T ' ) + 1 , γ t' (T ' ) ≤ γ d' (T ' ) . This holds the claim and therefore T

set

edge dominating set of T , γ t

'

which gives a contradiction to the fact that

can be obtained from

(T ) ≥ γ (T ) − 2 . Further since e '

' t

b

T ' by type-4 operation.

∉ S , S I E (T ' ) is a double edge dominating

γ t' (T ' ) ≥ γ t' (T ) − 2 = γ d' (T ) − 2 = S I E (T ' ) ≥ γ d' (T ' ) .

'

Hence

T ' ∈ℜ . Let D ' be any γ t' - set of T ' . By above claim, ex ∉ D ' . Since D ' U {ex , eb } is a total

Now we prove that

set of T ,

T '.

of

γ t' (T ' ) = γ d' (T ' ) ,

which implies that

Therefore by Theorem 1, we get

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

{

T ' =T − Uni=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

( { })

D' U Uni=1 ei ,ewi

is a total edge dominating set of T and hence.

component P3 = ea eb , we can choose set of

T

'

T '∈ℜ . Suppose D ' ⊂ E (T ' ) be a γ t' - set of T ' . Then

and hence

S⊆ E(T)

γ d' (T ) = S = 2n +

as a

γ d'

γ t' (T ' ) ≥ γ t' (T ) − 2n .

- set of T containing ex . Then

Since

T − ex has a

S I E(T ' ) is a γ d'

-

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 .

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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

γ d'

be a

P2 of S ( ex ) by ex' and ex'' in T . Let T ' = T − ex'' . Clearly, γ t' (T ' ) = γ t' (T ) . Let S

- set of T containing

edge

dominating

{e

w1

}

, ew2 ,..., ewn . By Observation 2, {ex' , ex'' } ⊆ S . Since S I E(T ) is a double

set

'

T'

of

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 Now by definition of

γ t'

γ d' (T ' ) + 2(n − 1) = γ d' (T )

'

{

=T − Uni=1 eui ,ei ,ewi

- set and . Hence

γ d' -

} . Then T is obtained from T

set, it is easy to observe that γ t (T '

γ t' (T ' ) = γ d' (T ' ) and T ' ∈ ℑ .

'

'

by type - 3 operation.

) + 2 (n − 1) = γ t' (T ) and

Applying the inductive hypothesis on

T ',

T ' ∈ℜ and hence T ∈ ℜ .

Sub case 4.2: For n = 1 . Denote the component component

P2 of S ( ex ) by ex' in T . Suppose S ( e y ) − S [ ex ] has a

H∈{P3, P4 , P6} in T , then T ' = T − S [ ex ] . Therefore we can easily check that T

T ' by type-6 operation. Now by definition of

γ d' -

set,

γ d' (T ' ) + 3 = γ d' (T ) .

For any

γ t'

is obtained from '

- 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 of

' S ( ey ) in T . By above discussion, S ( ey ) consists of a component P6 = eu e ew ex ex and g number of

components of P2 , denoted by that

S ( ey ) − S [ ex ] has no components P3 , P4 or P6 . Then we consider the structure

{e , e ,..., e } . Assume l = 2 . Then, let T 1

2

g

γ t' (T ' ) + 4 ≥ γ d' (T ) = γ d' (T ' ) + 5 ,

'

= T − S ey  . It can be easily checked

which is a contradiction to the fact that

g ≤ 1. 84

γ t' (T ' ) ≤ γ d' (T ' ) .

Hence


Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 Suppose

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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 } Therefore γ t

'

is

a

double

edge

dominating

set

of

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 If g = 0 , then

T ' . Hence T ' ∈ ℑ . Applying inductive hypothesis to T ' , T ' ∈ ℜ .

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

ez ∈S ,

degT ( ey ) = 3 . Since ex ∉S to double edge dominate ey , we have ey ∉S

by the selection of S . Therefore

and

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.

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Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012

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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.

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Characterization of Trees with Equal Total Edge Domination and Double Edge Domination Numbers