0. Domination in generalized Petersen graphs
Sheng-Chyang Liaw
廖 勝 強
Department of Mathematics
National Central University
Jhongli 32001, Taiwan
*Joint work with Yuan-Shin Li(李元馨)

1. Dominating set: A vertex subset S of a graph G is a dominating set
if each vertex in V(G) − S is adjacent to at least one
vertex in S.
The domination number of G is the cardinality of a
minimum dominating set of G, denoted by C5

2. Independent dominating set:
S is an independent dominating set if S is not only a dominating set but also an independent set.
The independent domination number of G is the cardinality of a minimum independent dominating set of G, denoted by C5

3. Total dominating set:
S is a total dominating set if each vertex v of G is adjacent to some vertex of S.
The total domination number of G is the cardinality
of a minimum total dominating set of G, denoted by . C5

4. Proposition:
γ(G) ≤ γi (G).
γ(G) ≤ γt (G).
If G is k-regular then

5. Generalized Petersen graph:
A generalized Petersen graph P(n, k)
vertex set: ={v1, v2, ..., vn} {u1, u2, ..., un}
edge set:
with vn+1 = v1, un+1= u1, 1 ≤ i ≤ n, and 1 ≤ k ≤

6. V={v1, v2, ..., vn} U={u1, u2, ..., un} E= {vivi+1, viui, uiui+k}
1 ≤ i ≤ n, and 1 ≤ k ≤ u4
v10 v4
u10 u3
u11
v11 v3 u2
u12 u1
u13
v12 v2
v13 v1
Generlized Petersen graph P(13,3)

7. Petersen graph = P(5,2):

8. Previous results:
γ(P(2k+1,k)) ≤ : A. Behzad, M. Behzad, and C. E. Praeger, On the domination number of the generalized Petersen graphs, Discrete Mathematics 308(2008), 603~610.
γ(P(2k+1,k))  : H. Yan, L. Kang, and G. Xu, The exact domination number of the generalized Petersen graphs, Discrete Mathematics 309(2009), 2596~2607. 8 8

9. Previous results:
γ(P(n,k)), k=1,2,3 : B. J. Ebrahimi, N. Jahanbakht, and E. S. Mahmoodian, Vertex domination of generalized Petersen graphs, Discrete Mathematics 309(2009), 4355~4361.
γt(P(n,2)) : J. Cao, W. Lin, and M. Shi, Total domination number of generalized Petersen graphs, Intelligent Information Management 2009, 15~18. 9 9

10. P(2k + 1, k) ,n ≡ 0,2(mod 3) P(2k, k) P(n, 1) , n≡2 (mod 4)
domination number
independent domination number
total domination number
P(2k + 1, k)
,n ≡ 0,2(mod 3)
, n ≡1(mod3)
P(2k, k)
P(n, 1)
, n≡2 (mod 4)
,n≡0,1,3 (mod 4)
,n ≡ 1,2 (mod 4)
, n ≡ 0,3 (mod 4)
, n≡4 (mod 6)
P(n, 2)
, n ≡ 1(mod3)
P(n, 3)
n ≡ 2,3 (mod 4)
, and n ≠ 11
n ≡ 0,1 (mod 4)
, or n=11
,n ≡ 2,3 (mod 4)
, n ≡ 0,1 (mod 4)
, n≡6,8 (mod 10)
, otherwise 10

11. γi(P(2k + 1, k)) = : Since γ(G) ≤ γi(G), we have γi(P(2k + 1, k)) ≥ .u1 u8 u7 u6 u5 u4 u3 u2 u9
u15
u12
u13
u14
u10
u11 v1 v2 v3 v4 v5 v6 v7 v8
v10 v9
v11
v12
v13
v14
v15
n = 5a
S0 = {V5t+1|0 ≤ t < a}
S1 = {U5t+3|0 ≤ t < a} ∪
{U5t+4|0 ≤ t < a};
|S0| + |S1| = a + 2a = 3a =

12. n = 5a + 3 S0 = {V5t+1|0 ≤ t < } ∪ {V5t| ≤ t ≤ a};
S1 = {U5t+3|0 ≤ t < a} ∪ {U5t+4|0 ≤ t < } ∪ {U5t+2| ≤ t ≤ a};
|S0| + |S1| = a a + a + 1 = 3a + 2 = u1 u3 u2 u4 u5 u6 u7 u8 u9
u10
u11
u12
u13 v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
n = 5a + 3 12 12

13. Total domination: If G is k-regular then .
a connected set D of a 3-regular graph G can dominate at most 2|D|+2 vertices in G with |D| ≥ 2 .
Proof. By induction on |D|.

14. Total domination in a 3-regular and 2n vertices graph:
If n = 3a + 1 and γt= |S| = , then we have the following two properties.
For any connected subset D ⊂ S, |D| ≤ 3.
S is formed by the union of exactly one 3-element connected set and a − 1 2-element connected sets for n = 3a + 1. 14 14

15. γ(P(2k,k))  :
For i = 1, ..., n, we define Si to be an induced subgraph of G. Si
P(10,5)

16. γ(P(2k,k))  : Let D be a minimum dominating set of G = P(2k, k).
vi+1 vi
ui+k+1
vi-1
ui-1 ui
ui+1
ui+k
ui+k-1
vi+k-1
vi+k
vi+k+1
Let D be a minimum dominating set of G = P(2k, k).
Therefore we know Si
We have
And each vertex of G belongs to exactly 6 Si
So, 16
Hence 6|D| ≥ 4n and 16

17. γi(P(2k,k))  : (1) n = 3a S0 = {v3t+1|0 ≤ t < a}
u12
u11
u10 u9 u8 u7 u6 u5 u4 u3 u2 u1
(1) n = 3a
S0 = {v3t+1|0 ≤ t < a}
S1 = {u3t+2|0 ≤ t < } ∪ {u3t|1 ≤ t ≤ }
|S0| + |S1| = a + a = 2a = 17 17

18. γi(P(2k,k))  : (2) n = 3a + 1 S0 = {v3t+1|0 ≤ t ≤ }u7 u8 u9
u10 u6 u5 u4 u3 u2 u1
(2) n = 3a + 1
S0 = {v3t+1|0 ≤ t ≤ }
∪ {v3t| < t ≤ a}
S1 = {u3t+2|0 ≤ t < a}
|S0| + |S1| = 2a + 1 = 18 18

19. γt(P(2k,k)) = n: An obviously upper bound of γt(P(2k, k)) is n.
Let D be the minimum total dominating set of G = P(2k, k).
Define Si={vi,vi+k,ui,ui+k} for i = 1, ..., n vi ui
ui+k
vi+k Si

20. γt(P(2k,k)) = n: Let Di = D ∩ Sivi ui
ui+k
vi+k
Let Di = D ∩ Si
Therefore |Di| ≥ 2.
each vertex of G belongs to exactly 2 Si Si
So,
Hence 2|D| ≥ 2n and |D| ≥ n 20 20

21. P(2k + 1, k) ,n ≡ 0,2(mod 3) P(2k, k) P(n, 1) , n≡2 (mod 4)
domination number
independent domination number
total domination number
P(2k + 1, k)
,n ≡ 0,2(mod 3)
, n ≡1(mod3)
P(2k, k)
P(n, 1)
, n≡2 (mod 4)
,n≡0,1,3 (mod 4)
,n ≡ 1,2 (mod 4)
, n ≡ 0,3 (mod 4)
, n≡4 (mod 6)
P(n, 2)
, n ≡ 1(mod3)
P(n, 3)
n ≡ 2,3 (mod 4)
, and n ≠ 11
n ≡ 0,1 (mod 4)
, or n=11
,n ≡ 2,3 (mod 4)
, n ≡ 0,1 (mod 4)
, n≡6,8 (mod 10)
, otherwise 21

22. E N D