Upload
alfred-walsh
View
285
Download
3
Embed Size (px)
Citation preview
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.
THANK YOU!