CHARACTERIZATION OF END
DOMINATION IN TREES
M. H. MUDDEBIHAL1
Department of Mathematics, Gulbarga Unviversity, Gulbarga-585106 Karnataka, INDIA
E-mail: mhmuddebihal@yahoo.co.in
A. R. SEDAMKAR2
Department of Mathematics, Gulbarga Unviversity, Gulbarga-585106 Karnataka, INDIA
E-mail:anil.sedamkar@gmail.com
ABSTRACT:
Let
G
=
(
V E
,
)
be a graph. A dominating setD
⊆
V
is an end dominating set ifD
contains all the endvertices in
V G
( )
. The end domination number ofG
is the minimum cardinality of an end dominating set ofG
. In this paper we provide a constructive characterization of the extremal treesT
with equal enddomination and
3
p
wherep
is the number of vertices inG
.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
|
andq
=
|
E
|
denote the number of vertices and edges of agraph
G
, respectively.In general, we use
X
to denote the subgraph induced by the set of verticesX
andN v
( )
and[ ]
N v
denote the open and closed neighborhoods of a vertexv
, respectively. The minimum (maximum) degreeamong the vertices of
G
is denoted byδ
( )
G
(
Δ
( )
G
)
. A vertex of degree one is called an end vertex and its neighbor is called support vertex. Moreover, the notationP
p will denote the path withp
vertices.A set
D
⊆
V
is a dominating set if every vertex not inD
is adjacent to at least one vertex inD
. The domination number ofG
, denoted byγ
( )
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 ofG
, ifD
contains all the end vertices. The end domination number of a graphG
, denoted byγ
e( )
G
, equals minimum cardinality of an end dominating set ofG
. 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 dominationand
3
p
RESULTS:
Theorem 1:For any path
P
p withp
≥
3
vertices,(
)
3
e p
p
P
p
γ
≤ −
.Proof:For
p
=
2
,(
)
3
e p
p
P
p
γ
−
. Now forp
≥
3
, letD
be an end dominating set ofP
p whose vertexset is
V P
( )
p=
{ ,
v v
1 2,
…
,
v
p}
. Note thatv v
1,
p∈
D
. Further any vertex ofV
−
D
is of order exactly two. Each vertex inV
−
D
is adjacent to a vertex inD
and to a vertex inV
−
D
. Suppose there aren
suchvertices, then
2n n
+ ≤
p
and so3
p
n
≤
. Thus|
|
3
p
D
= − ≤ −
p
n
p
and hence e( )
p3
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 treesT
withp
≤
10
. Suppose, therefore, that the result is true for all trees of order less thanp
wherep
≥
8
. Letγ
e( )
T
be the minimal enddominating set of
T
. We now show that3
e
p
γ
≥
. Among all the trees inℑ
. LetT
be chosen so that thesum
s T
( )
of the degrees of its vertices of degree at least one is minimum. With respect to this, letT
be chosensuch that the number of end vertices of
T
is minimum. Ifs T
( )
=
2
, thenT
≅
P
p and by Proposition [1],( )
3
e e
p
T
γ
=
γ
≥
. Suppose therefore thats T
( )
≥
3
, then there exist at least one vertexv
suchthat
deg( )
v
≥
3
. LetD
be aγ
e( )
T
-set ofT
.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 ofD
iii)
deg( )
v
=
3
Proof: i) Suppose
v
∈
D
. Then there existsa b c
, ,
∈
N v
( )
such thata b
,
∈
D
. LetT
'
be obtained fromT
by deleting either
a
orb
. ThenD
is an end dominating set ofT
'
, and so by definition ofγ
e, wehave
γ
e( ')
T
≤ ≤
γ
e|
D
|
=
γ
e. HenceT
'
∈ ℑ
. However asT
'
has fewer end vertices thanT
, we obtain aii) Thus assume
v
∉
D
and letb c
,
∈
N v
( )
such thatc
∉
D
andb
∈
D
. Ifd
∈
N v
( ) { , }
−
b c
is inD
, thenby deleting
b
ofT
, we obtain a contradiction as before. We therefore assume thatb
is the only vertex inD
which is adjacent to
v
.iii) Suppose
deg( )
v
≥
4
, let{
v v
1,
2,
…
,
v
n−2}
=
N v
( )
−
{
b c
,
}
, letu
=
v
1 and letw
be an end vertex of thecomponent of
T
−
v
that containsb
. Further letT
′
be the tree which arises fromT
by deleting the vertices,
1, 2,
,
2
i
v
i
=
… −
n
and joiningu
tow v v
,
2, ,
3…
,
v
n−2. Notethat
deg ( )
T′v
=
deg ( )
T′w
=
2
,deg ( )
T′u
=
deg( ) deg( ) 3
u
+
v
− ≥
deg( ) 1 3
u
+ ≥
, while all other verticeshave the same degree in
T
′
as inT
. Ifdeg( )
u
=
2
, thens T
( )
′ =
s T
( ) deg( ) deg ( )
−
v
+
T′u
=
s T
( ) 1
−
. Onthe other hand, if
deg( )
u
≥
3
, thens T
( )
′ =
s T
( ) deg( ) 3
−
v
− =
s T
( ) 3
−
. ThenD
is an end dominating setof
T
′
. AsT
′∈ ℑ
ands T
( )
′ <
s T
( )
, we obtain a contradiction in both cases. Thusdeg( )
v
=
3
.Remark 1: For any tree
T
of orderp
≥
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 ofv
inD
. By Claim 1,deg( )
b
=
1
. Ifdeg( )
c
=
3
, then by Claim 1,c
is adjacent to a vertex inV
−
D
other thanv
and to a vertex inD
. SupposeT
′
be obtained fromT
by deletingv b
,
. ThenD
is an end dominating set ofT
′
, and so by definitionof
γ
e, we have thatγ
e( )
T
′ ≤
γ
e( ) |
T
≤
D
|
=
γ
e. HenceT
′∈ℑ
. HoweverT
′
has fewer end vertices thanT
, 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 inD
,deg( )
b
=
1
by Remark 1. Letb
′
be the vertex adjacent tob
. Supposeb
′
is not an end vertex. Then by Claim 1,deg( )
b
′ =
2
. Letb
′′
be the neighbor ofb
′
different fromb
. ThenD
is an end dominating set of treeT
′
obtained fromT
by deleting the vertexb
. ThusT
′∈ℑ
andb
′
is an end vertex ofT
′
. Hence we may assume thatb
′
is an end vertex ofT
.We are now in a position to prove the remaining part of Theorem 2.
By Claim 2,
deg( )
a
=
deg( ) 1
b
=
. Supposea b
′ ′
,
be the neighbors ofa b
,
respectively which aredifferent from
v
. Necessarily,a b
′ ′∈
,
D
. Thendeg( )
a
′ =
deg( ) 1
b
′ =
by Claim 1 and Remark 1. As eachvertex in
D
is adjacent to no other vertex inD
, we may assume thata b
′ ′
,
as end vertices ofT
.If
p
=
11
, then( )
4
3
e
p
T
γ
= =
. Suppose, therefore thatp
≥
12
andT
′
be the componentof
T
−
{ , }
a a
′
. ThenD
∩
V T
( )
′
is an end dominating set ofT
′
, so that|
D
∩
V T
( ) |
′ ≥
γ
e( )
T
′
. Hence|
D
| 2
≥ +
γ
e( )
T
′
. Applying the inductive hypothesis to the treeT
′
of orderp
−
9
, we have2
( )
3
e
p
T
γ
′ ≥
−
and so e( ) |
|
3
p
T
D
γ
=
≥
Type 1: Join an end path
P
1 to an end vertexv
inT
.Type 2: Join an end path
P
2 to an end vertexv
inT
.Let
ℜ
be the class of all trees obtained by a finite sequence of operations type 1 and type 2. We will show thatT
∈ℜ
if and only ifT
∈ ℑ
.Lemma 1: Let
T
′∈ℑ
be a tree of orderp
. IfT
is obtained fromT
′
by the operation type 1, thenT
∈ ℑ
.Proof: Let
D
be aγ
e -set ofT
′
throughout the proof of this result andu
be an end vertex ofT
′
. SupposeT
is formed by attaching the singletonv
tou
. ThenD
∪
{ }
v
is an end dominating set ofT
and so1
( )
1
3
e3
p
p
T
γ
+
≤
≤
+
. Therefore e( )
3
p
T
γ
=
. ThusT
∈ ℑ
.Lemma 2: Let
T
′∈ℑ
be a tree of orderp
. IfT
is obtained fromT
′
by the operation type 2, thenT
∈ ℑ
.Proof: Let
D
be aγ
e -set ofT
′
. Supposev
is an end vertex ofT
′
. Thenv
∈
D
. LetT
be a tree which isobtained from
T
′
by adding the pathvxy
tov
. ThenD
∪
{ }
y
is an end dominating set ofT
and so2
( )
2
3
e3
p
p
T
γ
+
≤
≤
+
. Therefore e( )
3
p
T
γ
=
and soT
∈ ℑ
.We are now in a position to prove the main result of this section.
Theorem 3:
T
is inℜ
if any only ifT
is inℑ
.Proof: Assume
T
∈ℜ
. We show thatT
∈ ℑ
by using the induction ono T
( )
, the number of operations required to construct the treeT
. Ifo T
( )
=
0
, thenT
=
P
4, which is inℑ
. Assume then, for all treesT
′∈ℜ
with
o T
( )
′ <
k
wherek
≥
1
is an integer, thatT
′
is inℑ
. LetT
∈ℜ
be a tree witho T
( )
=
k
. ThenT
is obtained fromT
′
by one of the operations type 1 or type 2. But thenT
′∈ℜ
ando T
( )
′ <
k
. Applying the inductive hypothesis toT T
′ ′
,
is inℑ
. Hence by Lemma 1 or Lemma 2,T
is inℑ
.Conversely, we show that
T
∈ℜ
for a non trivial treeT
∈ ℑ
. We use induction onp
, the order of the treeT
. Ifp
=
2
, thenT
≅ ∈ℜ
P
2 . LetT
∈ ℑ
be a tree of orderp
≥
3
, and assume for all treesT
′∈ ℑ
of order
2
≤
p T
( )
′ <
p
, thatT
′∈ℜ
. Sincep
≥
3
inT diam T
′
,
( )
≥
2
. Ifdiam T
( )
=
2
, thenT
is a star with exactly 2 end vertices which can be constructed fromP
2 by applying the operation type 1. ThusT
∈ℜ
.Throughout the proof of this result
D
′
will be used to denote aγ
e -set of T.We give following two Claims for the next part of Theorem 3.
Claim 3: If
u
is an end vertex ofT
andv
is either another end vertex or the support vertex adjacent tou
, thenD
= ′ −
D
{ , }
u v
is not an end dominating set ofT
= ′ − −
T
u
v
.Proof: Suppose to the contrary that
D
′
is an end dominating set ofT
. Then2 2
( )
2
3
e3
p
p
T
γ
− +
≤
≤
−
. Thus3
2
3
p
p
+ ≤
, this yields a contradiction.Proof: Assume
D
′ −
{ }
u
is an end dominating set ofT
′
. Then1
( )
1
3
e3
p
p
T
γ
−
≤
′ ≤
−
. This yields acontradiction. Hence
( )
( ) 1
3
3
e
p
p T
T
γ
′ =
=
′ +
. ThusT
′∈ℑ
withp T
( )
′ = −
p
1
. By the inductionhypothesis,
T
′∈ℜ
. The treeT
can be constructed fromT
′
by applying the operation type 1. HenceT
∈ℜ
.Remark 2: Claim 4 implies that, if
i)
vxz
is an end path ofT
, then we may assume thatv
∉ ′
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 vertexr
of a longest path. Letv
be any vertex on a longest path at distance( ) 2
diam T
−
fromr
. Supposev
lies on an end pathvuw
. Then by (i) of Remark 2,v
∉ ′
D
, which implies thatv
is not adjacent to end vertex. Ifv
also lies on end pathvxz
, thenD
= ′ −
D
{ , }
x z
is anend 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 distancediam T
( ) 2
−
ordiam T
( ) 1
−
fromr
has degree at most 2.Suppose
v
be any vertex on a longest path at distancediam T
( ) 3
−
fromr
. Letvx x z
1 2′
be an endpath of
T
. Thenx
1∉ ′
D
and sov
∉ ′
D
, which means all neighbors ofv
have degree at least two.Assume
v
lies on an end pathvxz
, wherez
is an end vertex. Then, since each support vertex is adjacent to exactly one end vertex,vxz
is an end path. Ifv
is dominated by a vertex other thanx
, then{ , }
D
= ′ −
D
x z
is an end dominating set ofT
= ′ − −
T
x
z
, which is a contradiction by Claim 3. Hencev
is dominated byx
. ThusD
= ′ −
D
{ , }
x z
2′
is an end dominating set ofT
= −
T
′
x
1−
x
2−
z
′
andso
3
( )
2
3
e3
p
p
T
γ
−
≤
′ ≤
−
. This yields a contradiction. Hence1
( )
( )
3
3
e
p
p T
T
γ
′ =
−
=
′
. ThusT
′∈ℑ
. By induction assumptionT
′∈ℜ
. The treeT
can now be constructed fromT
′
by applying the operation type-2. HenceT
∈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.