Outer-connected domination numbers of block graphs

杜國豪指導教授：郭大衛教授

國立東華大學應用數學系碩士班

Outline:Introduction

Main result

• Full k-ary tree

• Block graph

Reference

Definition: For a graph a set is a dominating

set if . A dominating set is an outer-connected

dominating set(OCD set) if the subgraph induced by is connected.

Example:

( )S V G[ ] ( )N S V G

S

\V S

,G

Definition: For a graph a set is a dominating

set if . A dominating set is an outer-connected

dominating set(OCD set) if the subgraph induced by is connected.

Example:

( )S V G,G[ ] ( )N S V G

\V S

S

Definition:A full -ary tree with height denoted is a k-ary tree with all leaves are at same level.

k h ,k hT

k

3,2T

Proposition 1:If is a tree and is an outer-connected dominating set of , then either or every leaf of belongs to

Lemma 2: If is a cut-vertex of and are the components of then for every outer-connected dominating set of which contains there exists such that

T ST | | 1S n

v T .S

v G 1 2, , ,

kG G G

\ ,G vGS

,v ,i{ .( ( )) }

jj i

V G v S

Theorem 3: For all , 1h

1

2, 1

(2 1) , 1,2;( )

(2 1) (2 3), 3.

h

h hc

h if hT

h if h

2,2

( ) 5c

T 2,4

( ) 31 5 26c

T

Theorem 4: For all 3, 1,k h

1

,

1 ( 1) 1( ) .

1 ( 1) 1

h h

k hc

k kT

k k

3,3

( ) 40 7 33c

T

Definition:A block of a graph is a maximal -connected subgraph of A block graph is a graph which every block is a complete graph.The block-cut-vertex tree of a graph is a bipartite graph in which one partite set consists of the cut-vertices of , and the other has a vertex for each block of And adjacent to , if containing in

G.G

2

GH

Gi

bi

B .Gx

ib

iB x .G

Example:

Example:

Example:

Example:

Example: Red: cut-vertexBlue: block

Example:

r

Example:v

G

vr

Algorithm for block graphs:

min{| |: is an outer-connected dominating set of

wh cich ontains },vG v

a S S G

v

min{| |: is an outer-connected dominating set of

does n which },ot containvG v

b S S G

v

dominating \{min{| |: is a set of , ,

and \ is connected}.

}vG v

v

c S S v S

G S

G v

( ) min{ , }.r rG Gc

G a b

* * * *min{min{ },min{ 1},i i i iv v B B B Bi i

a n a n c n

* * * *

1

* * *

1

( ) , if 1,

min{ }, if 0,

i i i i

j i i

l

v B B B B vi

lv

B B B vij

I c e I e Ib

e d e I

*

1

, if 1,

, if 0,j

v v

lv

B vj

b Ic

e I

*

1

1 ,i

l

v Bi

n l n

*

1

.i

l

v Bi

I I

* * min{ },i iB B u ui

a n a n *

1

,i

l

B ui

b b

*

1

,i

l

B ui

c c

* * min{ },

i iB B u uid c n c

* * *min{ , },B B Bi

e b d* | ( ) |,B

m V B* *

1

,i

l

B B ui

n m l n

*

*

*

1, if 1,

0, if 1.B

B

B

m lI

m l

Initial values:

Time complexity:Each vertex uses a constant time for computing its parameters, the time complexity of this algorithm is

* * * * *

* * *

1, , 0, 1, 1,

| ( ) |, | ( ) |, 1.B B B B B

B B B

a b c d e

m V B n V B I

.( )n

Example 1:

Example 1:

1, ,0,1,1,3,3,1 1, ,0,1,1,3,3,1 1, ,0,1,1,3,3,1 1, ,0,1,1,3,3,1 1, ,0,1,1,2,2,1 1, ,0,1,1,2,2,1 1, ,0,1,1,4,4,1

3,2,2,5,2 1,1,1,3,1 1,1,1,2,1 2,2,2,4,2 1,1,1,4,1

3,2,2,5,2,2,6,0 2,1,1,3,1,2,4,0 8,4,4,5,4,4,11,0

13,8,7,19,0

( , , , , )a b c n I* * * * * * * *( , , , , , , , )a b c d e m n I

Example 1:* * *( , , )a c n

* * * *min{min{ },min{ 1},i i i iv v B B B Bi i

a n a n c n

19 min(min( 3, 2, 3),min( 3, 2, 6))a

13

(3,2,6) (2,1,4) (8,4,11)

a

Example 1:13a

8 10

min{| |: is an OCD set of

which conta },insvG v

a S S G

v

Example 1:* * *( , , )d e I

* * *

1

min{ }, if 0,j i i

l

v B B B vij

b e d e I

7 min(3,2,1)b

8

(5,2,0) (3,1,0) (5,4,0)

b

Example 1:8b

35 2 44

( ) 8c

G

min{| |: is an OCD set of

whic does not coh }nta n ,ivG v

b S S G

v

Example 1:

( ) 8c

G

Example 2:

2,4T

Example 2:

(26,27)( , )a b

Red: cut-vertexBlue: block

Example 2:

2,4

( ) 26c

T

Example 2:

| | 27S

r

Example 3:

Red: cut-vertexBlue: block

r

Example 3:

Example 3:r

( , , , , )a b c n I* * * * * * * *( , , , , , , , )a b c d e m n I

(1, ,0,1,1,3,3,1)(1, ,0,1,1,4,4,1)

(1, ,0,1,1,2,2,1)(1, ,0,1,1,2,2,1)

(1, ,0,1,1,4,4,1)

(1, ,0,1,1,3,3,1)(1, ,0,1,1,4,4,1)

(1, ,0,1,1,2,2,1)

(1, ,0,1,1,2,2,1)

(1,1,1,3,1) (1,1,1,4,1)

(8,3,3,4,3,5,11,1) (7,2,2,3,2,6,10,1)

(12,6,6,7,6,4,19,0)(6,4,4,11,4,3,13,1)

(1,1,1,2,1) (1,1,1,2,1) (1,1,1,4,1)

(4,4,4,11,1) (3,3,3,10,1) (3,2,2,6,2) (1,1,1,2,1)

(20,12,12,32,2)

Example 3:

( ) 12c

G

Reference:Akhbari, R. Hasni, O. Favaron, H. Karami and S. M. Sheikholeslami, "On the outer-connected domination in graphs," J. Combin. Optimi. DOI 10.1007/s10878-011-9427-x (2011).J. Cyman, The outer-connected domination number of a graph, Australas. J. Combin., 38 (2007), 35-46.H. Jiang and E. Shan, Outer-connected domination number in graph, Utilitas Math., 81 (2010), 265-274.

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