CHARACTERIZATION OF END DOMINATION IN TREES

CHARACTERIZATION OF END

DOMINATION IN TREES

M. H. MUDDEBIHAL1

Department of Mathematics, Gulbarga Unviversity, Gulbarga-585106 Karnataka, INDIA

E-mail: [email protected]

A. R. SEDAMKAR2

Department of Mathematics, Gulbarga Unviversity, Gulbarga-585106 Karnataka, INDIA

E-mail:[email protected]

ABSTRACT:

Let

G

=

(

V E

,

)

be a graph. A dominating set

D

⊆

V

is an end dominating set if

D

contains all the end

vertices in

V G

( )

. The end domination number of

G

is the minimum cardinality of an end dominating set

of

G

. In this paper we provide a constructive characterization of the extremal trees

T

with equal end

domination and

3

p

 

 

 

where

p

is the number of vertices in

G

.

Key Words: Order / Degree / Dominating set / End dominating set / End domination number.

Subject Classification Number:AMS – 05C69, 05C70.

INTRODUCTION:

In this paper we follow the notations of

[ ]

1

. All the graphs considered here are simple, finite, non-trivial, undirected and connected. 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 subgraph 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 minimum (maximum) degree

among the vertices of

G

is denoted by

δ

( )

G

(

Δ

( )

G

)

. A vertex of degree one is called an end vertex and its neighbor is called support vertex. Moreover, the notation

P

_p will denote the path with

p

vertices.

A set

D

⊆

V

is a dominating set if every vertex not in

D

is adjacent to at least one vertex in

D

. The domination number of

G

, denoted by

γ

( )

G

, is the minimum cardinality of a dominating set. The concept of domination in graphs with its many variations of parameters is now well studied in graph theory (see [3] and [4]).

A dominating set

D

is called an end dominating set of

G

, if

D

contains all the end vertices. The end domination number of a graph

G

, denoted by

γ

_e

( )

G

, equals minimum cardinality of an end dominating set of

G

. End domination in graphs was introduced by J.H. Hattingh and M.A. Henning [2] and is now well studied in graph theory.

In this paper we provide a constructive characterization of extremal trees

T

with equal end domination

and

3

p

 

 

RESULTS:

Theorem 1:For any path

P

_p with

p

≥

3

vertices,

(

)

3

e p

p

P

p

γ

≤

−

 

_{ }

 

.

Proof:For

p

=

2

,

(

)

3

e p

p

P

p

γ

−

 

_{ }

 



. Now for

p

≥

3

, let

D

be an end dominating set of

P

_p whose vertex

set is

V P

( )

_p

=

{ ,

v v

_{1 2}

,

…

,

v

_p

}

. Note that

v v

₁

,

_p

∈

D

. Further any vertex of

V

−

D

is of order exactly two. Each vertex in

V

−

D

is adjacent to a vertex in

D

and to a vertex in

V

−

D

. Suppose there are

n

such

vertices, then

2n n

+ ≤

p

and so

3

p

n

≤

 

_{ }

 

. Thus

|

|

3

p

D

=

p

− ≤

n

p

−

 

_{ }

 

and hence e

( )

p

3

p

P

p

γ

≤

−

 

_{ }

 

.

To prove the next result we define the following family as

Extremal trees

/ is a tree of order such that ( )

3

e

p

T T

p

γ

T



 



ℑ =

_

=

_{ }

_

 





.

Theorem 2:For any tree

T

,

( )

3

e

p

T

γ

≥

 

_{ }

 

Proof:We use induction on

p

. It is easy to check that, the result is true for all trees

T

with

p

≤

10

. Suppose, therefore, that the result is true for all trees of order less than

p

where

p

≥

8

. Let

γ

_e

( )

T

be the minimal end

dominating set of

T

. We now show that

3

e

p

γ

≥

 

_{ }

 

. Among all the trees in

ℑ

. Let

T

be chosen so that the

sum

s T

( )

of the degrees of its vertices of degree at least one is minimum. With respect to this, let

T

be chosen

such that the number of end vertices of

T

is minimum. If

s T

( )

=

2

, then

T

≅

P

_p and by Proposition [1],

( )

3

e e

p

T

γ

=

γ

≥

 

_{ }

 

. Suppose therefore that

s T

( )

≥

3

, then there exist at least one vertex

v

such

that

deg( )

v

≥

3

. Let

D

be a

γ

_e

( )

T

-set of

T

.

We give following two Claims for the next part of Theorem 2.

Claim 1:If

v

is a vertex of degree at least 3, then i)

v

∉

D

ii)

v

is adjacent to exactly one vertex of

D

iii)

deg( )

v

=

3

Proof: i) Suppose

v

∈

D

. Then there exists

a b c

, ,

∈

N v

( )

such that

a b

,

∈

D

. Let

T

'

be obtained from

T

by deleting either

a

or

b

. Then

D

is an end dominating set of

T

'

, and so by definition of

γ

_e, we

have

γ

_e

( ')

T

≤

γ

_e

≤

|

D

|

=

γ

_e. Hence

T

'

∈ ℑ

. However as

T

'

has fewer end vertices than

T

, we obtain a

ii) Thus assume

v

∉

D

and let

b c

,

∈

N v

( )

such that

c

∉

D

and

b

∈

D

. If

d

∈

N v

( ) { , }

−

b c

is in

D

, then

by deleting

b

of

T

, we obtain a contradiction as before. We therefore assume that

b

is the only vertex in

D

which is adjacent to

v

.

iii) Suppose

deg( )

v

≥

4

, let

{

v v

₁

,

₂

,

…

,

v

_n−2

}

=

N v

( )

−

{

b c

,

}

, let

u

=

v

₁ and let

w

be an end vertex of the

component of

T

−

v

that contains

b

. Further let

T

′

be the tree which arises from

T

by deleting the vertices

,

1, 2,

,

2

i

v

i

=

…

n

−

and joining

u

to

w v v

,

₂

, ,

₃

…

,

v

_n−2. Note

that

deg ( )

T′

v

=

deg ( )

T′

w

=

2

,

deg ( )

T′

u

=

deg( ) deg( ) 3

u

+

v

− ≥

deg( ) 1 3

u

+ ≥

, while all other vertices

have the same degree in

T

′

as in

T

. If

deg( )

u

=

2

, then

s T

( )

′ =

s T

( ) deg( ) deg ( )

−

v

+

_T′

u

=

s T

( ) 1

−

. On

the other hand, if

deg( )

u

≥

3

, then

s T

( )

′ =

s T

( ) deg( ) 3

−

v

− =

s T

( ) 3

−

. Then

D

is an end dominating set

of

T

′

. As

T

′ ∈ ℑ

and

s T

( )

′ <

s T

( )

, we obtain a contradiction in both cases. Thus

deg( )

v

=

3

.

Remark 1: For any tree

T

of order

p

≥

3

, no two vertices of degree one are adjacent.

Claim 2: No two vertices of degree 3 are adjacent in

D

.

Proof: Using the notation employed in Claim 1,

b

is the only neighbor of

v

in

D

. By Claim 1,

deg( ) 1

b

=

. If

deg( )

c

=

3

, then by Claim 1,

c

is adjacent to a vertex in

V

−

D

other than

v

and to a vertex in

D

. Suppose

T

′

be obtained from

T

by deleting

v b

,

. Then

D

is an end dominating set of

T

′

, and so by definition

of

γ

e, we have that

γ

e

( )

T

′ ≤

γ

e

( ) |

T

≤

D

|

=

γ

e. Hence

T

′∈ ℑ

. However

T

′

has fewer end vertices than

T

,

we obtain a contradiction.

Using the notation employed in the proof of Claim 1, the vertex

b

∈

D

and as it is not adjacent to any vertex in

D

,

deg( ) 1

b

=

by Remark 1. Let

b

′

be the vertex adjacent to

b

. Suppose

b

′

is not an end vertex. Then by Claim 1,

deg( )

b

′ =

2

. Let

b

′′

be the neighbor of

b

′

different from

b

. Then

D

is an end dominating set of tree

T

′

obtained from

T

by deleting the vertex

b

. Thus

T

′∈ ℑ

and

b

′

is an end vertex of

T

′

. Hence we may assume that

b

′

is an end vertex of

T

.

We are now in a position to prove the remaining part of Theorem 2.

By Claim 2,

deg( )

a

=

deg( ) 1

b

=

. Suppose

a b

′ ′

,

be the neighbors of

a b

,

respectively which are

different from

v

. Necessarily,

a b

′ ′∈

,

D

. Then

deg( )

a

′ =

deg( ) 1

b

′ =

by Claim 1 and Remark 1. As each

vertex in

D

is adjacent to no other vertex in

D

, we may assume that

a b

′ ′

,

as end vertices of

T

.

If

p

=

11

, then

( )

4

3

e

p

T

γ

= =

 

_{ }

 

. Suppose, therefore that

p

≥

12

and

T

′

be the component

of

T

−

{ , }

a a

′

. Then

D

∩

V T

( )

′

is an end dominating set of

T

′

, so that

|

D

∩

V T

( ) |

′ ≥

γ

_e

( )

T

′

. Hence

|

D

| 2

≥ +

γ

_e

( )

T

′

. Applying the inductive hypothesis to the tree

T

′

of order

p

−

9

, we have

2

( )

3

e

p

T

γ

′ ≥



_

−



_





and so e

( ) |

|

3

p

T

D

γ

=

≥

 

_{ }

Type 1: Join an end path

P

₁ to an end vertex

v

in

T

.

Type 2: Join an end path

P

₂ to an end vertex

v

in

T

.

Let

ℜ

be the class of all trees obtained by a finite sequence of operations type 1 and type 2. We will show that

T

∈ ℜ

if and only if

T

∈ ℑ

.

Lemma 1: Let

T

′∈ ℑ

be a tree of order

p

. If

T

is obtained from

T

′

by the operation type 1, then

T

∈ ℑ

.

Proof: Let

D

be a

γ

_e -set of

T

′

throughout the proof of this result and

u

be an end vertex of

T

′

. Suppose

T

is formed by attaching the singleton

v

to

u

. Then

D

∪

{ }

v

is an end dominating set of

T

and so

1

( )

1

3

e

3

p

p

T

γ

+





 

≤

≤

+





 





 

. Therefore e

( )

3

p

T

γ

=

 

_{ }

 

. Thus

T

∈ ℑ

.

Lemma 2: Let

T

′∈ ℑ

be a tree of order

p

. If

T

is obtained from

T

′

by the operation type 2, then

T

∈ ℑ

.

Proof: Let

D

be a

γ

_e -set of

T

′

. Suppose

v

is an end vertex of

T

′

. Then

v

∈

D

. Let

T

be a tree which is obtained from

T

′

by adding the path

vxy

to

v

. Then

D

∪

{ }

y

is an end dominating set of

T

and so

2

( )

2

3

e

3

p

p

T

γ

+





 

≤

≤

+





 





 

. Therefore e

( )

3

p

T

γ

=

 

_{ }

 

and so

T

∈ ℑ

.

We are now in a position to prove the main result of this section.

Theorem 3:

T

is in

ℜ

if any only if

T

is in

ℑ

.

Proof: Assume

T

∈ ℜ

. We show that

T

∈ ℑ

by using the induction on

o T

( )

, the number of operations required to construct the tree

T

. If

o T

( )

=

0

, then

T

=

P

₄, which is in

ℑ

. Assume then, for all trees

T

′ ∈ ℜ

with

o T

( )

′ <

k

where

k

≥

1

is an integer, that

T

′

is in

ℑ

. Let

T

∈ℜ

be a tree with

o T

( )

=

k

. Then

T

is obtained from

T

′

by one of the operations type 1 or type 2. But then

T

′∈ℜ

and

o T

( )

′ <

k

. Applying the inductive hypothesis to

T T

′

,

′

is in

ℑ

. Hence by Lemma 1 or Lemma 2,

T

is in

ℑ

.

Conversely, we show that

T

∈ ℜ

for a non trivial tree

T

∈ ℑ

. We use induction on

p

, the order of the tree

T

. If

p

=

2

, then

T

≅

P

₂

∈ ℜ

. Let

T

∈ ℑ

be a tree of order

p

≥

3

, and assume for all trees

T

′ ∈ ℑ

of order

2

≤

p T

( )

′ <

p

, that

T

′∈ℜ

. Since

p

≥

3

in

T diam T

′

,

( )

≥

2

. If

diam T

( )

=

2

, then

T

is a star with exactly 2 end vertices which can be constructed from

P

₂ by applying the operation type 1. Thus

T

∈ ℜ

. Throughout the proof of this result

D

′

will be used to denote a

γ

e -set of T.

We give following two Claims for the next part of Theorem 3.

Claim 3: If

u

is an end vertex of

T

and

v

is either another end vertex or the support vertex adjacent to

u

, then

D

= ′ −

D

{ , }

u v

is not an end dominating set of

T

= ′ − −

T

u

v

.

Proof: Suppose to the contrary that

D

′

is an end dominating set of

T

. Then

2 2

( )

2

3

e

3

p

p

T

γ

− +





 

≤

≤

−





 





 

. Thus

3

2

3

p

p

 

 

+ ≤

 

 

 

 

, this yields a contradiction.

Proof: Assume

D

′ −

{ }

u

is an end dominating set of

T

′

. Then

1

( )

1

3

e

3

p

p

T

γ

−





 

≤

′ ≤

−





 





 

. This yields a

contradiction. Hence

( )

( ) 1

3

3

e

p

p T

T

γ

′ =

  

_{  }

=

′ +



_

  



. Thus

T

′∈ ℑ

with

p T

( )

′ =

p

−

1

. By the induction

hypothesis,

T

′ ∈ ℜ

. The tree

T

can be constructed from

T

′

by applying the operation type 1. Hence

T

∈ ℜ

.

Remark 2: Claim 4 implies that, if

i)

vxz

is an end path of

T

, then we may assume that

v

∉ ′

D

, since otherwise the tree is constructible.

ii) Every support vertex of

T

is adjacent to exactly one end vertex, since otherwise it is constructible.

We are now in a position to prove the remaining part of Theorem 3.

Let

T

be rooted at an end vertex

r

of a longest path. Let

v

be any vertex on a longest path at distance

( ) 2

diam T

−

from

r

. Suppose

v

lies on an end path

vuw

. Then by (i) of Remark 2,

v

∉ ′

D

, which implies that

v

is not adjacent to end vertex. If

v

also lies on end path

vxz

, then

D

= ′ −

D

{ , }

x z

is an

end dominating set of

T

= ′ − −

T

x

z

, this is a contradiction by Claim 3. Thus we assume that each vertex on a longest path at distance

diam T

( ) 2

−

or

diam T

( ) 1

−

from

r

has degree at most 2.

Suppose

v

be any vertex on a longest path at distance

diam T

( ) 3

−

from

r

. Let

vx x z

_{1 2}

′

be an end

path of

T

. Then

x

₁

∉ ′

D

and so

v

∉ ′

D

, which means all neighbors of

v

have degree at least two.

Assume

v

lies on an end path

vxz

, where

z

is an end vertex. Then, since each support vertex is adjacent to exactly one end vertex,

vxz

is an end path. If

v

is dominated by a vertex other than

x

, then

{ , }

D

= ′ −

D

x z

is an end dominating set of

T

= ′ − −

T

x

z

, which is a contradiction by Claim 3. Hence

v

is dominated by

x

. Thus

D

= ′ −

D

{ , }

x z

₂

′

is an end dominating set of

T

=

T

′

−

x

₁

−

x

₂

−

z

′

and

so

3

( )

2

3

e

3

p

p

T

γ

−





 

≤

′ ≤

−





 





 

. This yields a contradiction. Hence

1

( )

( )

3

3

e

p

p T

T

γ

′ =

−

=



_

′



_





. Thus

T

′∈ ℑ

. By induction assumption

T

′∈ℜ

. The tree

T

can now be constructed from

T

′

by applying the operation type-2. Hence

T

∈ℜ

.

REFERENCES:

[1] F. Harary, Graph Theory, Adison Wesley, Reading Mass (1972).

[2] J.H. Hattingh and M.A. Henning, Characterization of trees with equal domination parameters, Journal of graph theory 34 (2000), 142-153.