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 G, a set S  V (G) is a dominating
set if N[S ]  V (G) .
 A dominating set S is an outer-connected
dominating set(OCD set) if the subgraph
induced by V \ S is connected.

Example:
Definition:
For a graph G, a set S  V (G) is a dominating
set if N[S ]  V (G) .
 A dominating set S is an outer-connected
dominating set(OCD set) if the subgraph
induced by V \ S is connected.

Example:
Definition:
full k-ary tree with height h denoted Tk ,his a
k-ary
k tree with all leaves are at same level.
A
T3,2
Proposition 1:
If T is a tree and S is an outer-connected
dominating set of T, then either | S | n  1
or
every leaf vof Tbelongs to S .
Lemma 2:
If v is a cut-vertex of G and G1 , G2 , , Gk are
the components of
Gthen
\ v, for every outerG
connected dominating set of Swhich
contains vthere
exists isuch
that
,
,
( V (G j )) {v}  S .
j i
Theorem 3: For all h  1 ,
if h  1,2;
(2  1)  h,
 c(T2,h )   h1
(2  1)  (2h  3), if h  3.
h 1
 (T )  5
c
2,2
 (T )  31  5  26
c
2,4
Theorem 4: For all k  3, h  1,
h 1
h
k  1 (k  1)  1

.
 c (Tk ,h ) 
k 1
(k  1)  1
 (T )  40  7  33
c
3,3
Definition:
A block
of a graph G is a maximal 2-connected
subgraph of G.
A block graph is a graph which every block is a
complete graph.
The block-cut-vertex tree of a graph Gis a
bipartite graph Hin which one partite set
consists of the cut-vertices of G
, and the other
.
has a vertex b
for
each block Bofi GAnd
bi
i
xin G.
adjacent to ,xif Bcontaining
i
Example:
Example:
Example:
Example:
Example:
Red: cut-vertex
Blue: block
Example:
r
Example:
Gv
r
v
Algorithm for block graphs:

aG  min{| S |: S is an outer-connected dominating set of Gv
v
which contains v},

bG  min{| S |: S is an outer-connected dominating set of Gv
v
which does not contain v},

cG  min{| S |: S is a dominating set of Gv \ {v}, v  S ,
v
and Gv \ S is connected}.

 (G)  min{a , b }.
c
Gr
Gr

*
*
*
*
av  nv  min{min{
a

n
},min{
c

n

1},
B
B
B
B
i
i
i
i
i
 I   (c*  e* ) I * e* , if I  1,
B
B
B
v
 v i 1 B
bv   l
*
*
*
d B  eB }, if I v  0,
 eB  min{
i
 j 1
if I v  1,
bv ,
l
cv   *
eB , if I v  0,


 j 1
l
*
nv  1  l   nB ,
l

i
j


l
i 1
Iv   I B .
i 1
*
i
i
i
j

i
i
i
i
i

aB*  nB*  min{
a

n
},
u
u
i
i

i
l
bB   bu ,
*
i 1
l

i
cB*   cu ,
i 1
*
 *
d B  cB  min{
nu  cu },
i
i
i



i
eB  min{
bB , d B },
i
*
*
*
mB* | V ( B) |,
l
nB  mB  l   nu ,
i 1
*

1, if mB  l  1,
*
IB  
*
0, if mB  l  1.
*
*
i
Initial values:

aB  1, bB  , cB  0, d B  1, eB  1,
*
*
*
*
*
mB | V ( B) |, nB | V ( B) |, I B  1.
*
*
*
Time complexity:
Each
vertex uses a constant time for computing
its parameters, the time complexity of this
algorithm is (n).
Example 1:
Example 1:
(a, b, c, n, I )
(a* , b* , c* , d * , e* , m* , n* , I * )
13,8,7,19,0
3, 2, 2,5, 2, 2,6,0
3, 2, 2,5, 2
 2,1,1,3,1, 2, 4,0
1,1,1,3,1
1,1,1, 2,1
 2, 2, 2, 4, 2 
8, 4, 4,5, 4, 4,11,0
1,1,1, 4,1
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, ,0,1,1,3,3,11, ,0,1,1, 2, 2,11, ,0,1,1, 4, 4,1
Example 1:
(a* , c* , n* )
a  13
(3,2,6)
(2,1,4)
(8,4,11)
*
*
*
*
av  nv  min{min{
a

n
},min{
c

n
 1},
B
B
B
B
i
i
i
i
i
i
a  19  min(min(3, 2, 3),min(3, 2, 6))
aG  min{| S |: S is an OCD set of Gv
Example 1:
v
a  13
8
which contains v},
10
Example 1:
(d * , e* , I * )
b8
(5,2,0)
(3,1,0)
(5,4,0)
l
*
*
bv   eB*  min{
d

e
}, if I v  0,
B
B
i
j 1
j
b  7  min(3,2,1)
i
i
bG  min{| S |: S is an OCD set of Gv
v
Example 1:
which does not contain v},
b 8
5
3
2
 (G)  8
c
4
4
Example 1:
 (G)  8
c
Example 2:
T2,4
Red: cut-vertex
Example 2:
Blue: block
(26,27) (a, b)
Example 2:
 (T )  26
c
2,4
Example 2:
| S | 27
Example 3:
r
Example 3:
Red: cut-vertex
Blue: block
r
(20,12,12,32,2)
r
Example 3:
(6,4,4,11,4,3,13,1)
(4,4,4,11,1)
(8,3,3,4,3,5,11,1)
(a, b, c, n, I )
(a* , b* , c* , d * , e* , m* , n* , I * )
(12,6,6,7,6,4,19,0)
(1, ,0,1,1,2,2,1)
(3,3,3,10,1)
(3,2,2,6,2)
(1,1,1,2,1)
(7,2,2,3,2,6,10,1)
(1, ,0,1,1,3,3,1)
(1, ,0,1,1,2,2,1)
(1, ,0,1,1,4,4,1)
(1,1,1,3,1) (1,1,1,4,1) (1,1,1,2,1)
(1,1,1,2,1)
(1,1,1,4,1)
(1, ,0,1,1,4,4,1)
(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)
Example 3:
 (G)  12
c
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!