1. Secure and Secure-dominating Set of Cartesian Product Graphs
Kai-Ping Huang and Justie Su-Tzu Juan
Department of Computer Science and Information Engineering
National Chi-Nan University

2. Outline Introduction Conclusions Secure set Secure-dominating set
Preliminary
Main result
Conclusions 2

3. Introduction Def: Let G = (V, E) be a graph. If v  V and S ⊆ V :
N[S]
Introduction
N(v) v S
Def:
Let G = (V, E) be a graph. If v  V and S ⊆ V :
1. N(v) ={u  V : vu  E}.
2. N[v]= N(v) ∪{v}.
3. N(S) =∪vSN(v).
4. N[S]= N(S) ∪ S.

4. A(u) ={1, 2}
A(v) ={3}
A(u) = {2}
A(v) = {1, 3}
Introduction G S 2 u
D(u) ={u}
D(v) ={v}
D(u) = {u, v}
D(v) = ∅ 1 v 3
Def:
5. A : S → Ƥ (V(G) − S) is called an attack on S (in G) if
A(u) ⊆ N(u) − S for all u  S and A(u) ∩ A(v) = ∅
for all u, v  S, u  v.
6. D : S → Ƥ (S) is called a defense of S if D(u) ⊆ N[u] ∩
S for all u  S and D(u) ∩ D(v) = ∅ for all u, v  S,
u  v.
即為在圖g上找了一個集合s，我們把跟s相鄰且不在s上的點稱為attack，則定義a(u)代表不在s集合內而跟u相鄰的攻擊函數，且規定當定義其為a(u)時則不能在定義其為a(v)
同理d為在s集合中，跟s相鄰且在s中的點稱為defense，定義d(u)為在s集合內而跟u相鄰的防禦函數，且規定當定義為d(u)時則不能在定義其為d(v) 4

5. Introduction S G u v 1 2 3
Def:
7. secure set : All attack A on S, there exists a defense of
S corresponding to A.
8. s(G) = min{|S| : S is a secure set of G}.
對所有在s上的攻擊a而言，都能找到對應攻擊a的有效防禦

6. Introduction Def: 9. Dominating set : G if N[S] = V(G).
10. Secure-dominating set : S is a secure set of G that is
also a dominating set of G.
11. γs(G) = min{|S| : S is a secure-dominating set of G}.

7. Secure-dominating set
Introduction
General graph G Pn
Pm × Pn Km
Pn1  Pn2  …  Pnk
Km1  Km2  … Kmk
Secure set
Brigham et al, 2007 [1]
The thesis
Secure-dominating set
Chang et al, 2008 [2]
[1] R. C. Brigham, R. D. Dutton, S. T. Hedetniemi, “Security in graphs,” Discrete Appl. Math., 155 (2007),
[2] Chia-Lang Chang, Tsui-Ping Chang, David Kuo, “Secure and secure-dominating set of graphs,” National Dong Hwa University Applied Mathematics, Manuscript.

8. Secure set - Preliminary
Proposition 1. [1]
If S is a secure set of G, then for each v in S, |N[v] ∩ S| ≥ |N(v) − S|.
Corollary 2. [1]
If S1 and S2 are vertex disjoint secure sets in the same graph, then S1 ∪ S2 is a secure set.

9. Secure set - Preliminary
Proposition 3. [1]
s(Pm × Pn) = min{m, n, 3}.
P3 × P2 P5 × P5
s(G) = 2 s(G) =3

10. Secure set - Main result
Theorem 4.
1 < n1  n2  …  nk1  nk
1. When n1 = n2 =2,
s(Pn1  P n2  …  Pnk)  4n3  …  nk2
2. When 2 < n2,
s(Pn1  P n2  …  Pnk)  3n1  n2  …  nk2

11. Secure set - Main result
s(Pn1 × Pn2 × Pn3), n1  n2  n3
P2 × P2 × P2 P2 × P3 × P P3 × P3 × P3
P2 × P2 × P3 P2 × P3 × P P3 × P3 × P4 … … …

12. Secure set - Main result
s(Pn1 × Pn2 × Pn3 ), n1  n2  n3
G = P4 × Pn2 × Pn3 , s(G)  12
G = P5 × Pn2 × Pn3 , s(G)  15
G = Pn1 × Pn2 × Pn3, s(G)  3n1 …

13. Secure set - Main result
Lemma 5.
1. When n1 = n2 =2, s(Pn1 Pn2 Pn3)  4
2. When 2 < n2, s(Pn1 Pn2 Pn3)  3n1
Pn1  Pn2  …  Pnk = (Pn1  Pn2  …  Pnk2 )  Pnk1  Pnk
1. When n1 = n2 =2,
s(Pn1  P n2  …  Pnk)  4n3  …  nk2
2. When 2 < n2,
s(Pn1  P n2  …  Pnk)  3n1  n2  …  nk2

14. Secure set - Main result
Theorem 6. [1]
s(Km) = K7 K4

15. Secure set - Main result
Theorem 7.
1.When mk1 is even,
2.When mk1 is odd,
Km1  K m2  … Kmk1  Kmk = (Km1  K m2  … Kmk1)  Kmk

16. Secure set - Main result
Km1  K m2
if m1 odd
even l a b
Km2
m2 − l c
Km1

17. Secure set - Main result
Lemma 8.
1.When m1 is even,
2.When m1 is odd,
Km1  K m2  … Kmk1  Kmk = (Km1  K m2  … Kmk1)  Kmk
1.When mk1 is even,
2.When mk1 is odd,

18. Secure-dominating set - Preliminary
Theorem 9. [2]
For any graph G with |V(G)| = n, γs(G) ≥ n/2.
Theorem 10. [2]
For all n ≥ 2, γs(Pn) = n/2.
Corollary 11. [2]
1. γs(G × Pn) ≤ n/2 |V(G)|.
2. When n is even：
γs(G × Pn) = n/2 |V(G)|.
V(Pn) = {v1,v2, ··· ,vn} , S = {(u, vi): u ∈ V(G), i ≡ 2, 3 (mod 4)} is a secure-dominating set.
P5 × P8

19. Secure-dominating set - Preliminary
Lemma 12. [2]
For all n ≥ 1, S = {(2, j):1 ≤ j ≤ n}∪{(3, j):1 ≤ j ≤ n,
j ≡ 1(mod 2)} is a secure-dominating set of P3 × Pn.
Lemma 13. [2]
For all n ≥ 1, S = {(i, j): i = 2, 4, 1 ≤ j ≤ n}∪{(3, j):1 ≤ j ≤ n,
j ≡ 1(mod 2)} is a secure-dominating set of P5 × Pn.
Theorem 14. [2]
For all m, n ≥ 2, γs(Pm × Pn) = mn/2.
P7 × P7 (P3  P4) × P7
P3 × P7

20. Secure-dominating set - Preliminary
wA(v2) = 0
Def:
wA(v) = 1 − |A(v)| for all v ∈ S.
Lemma 16. [2]
1. wA(v) ∈ {−1, 0, 1}.
= k ≥ 1, 1 ≤ i ≤ k.
3. Vertex disjoint paths Pi, wA(vi,1) = 1,wA(vi,li ) = −1, and wA(vi,j ) = 0 for all i, j, 1 ≤ i ≤ k, 2 ≤ j ≤ li − 1.
There exists a defense D of S corresponding to A.
wA(v1) = 1
wA(v3) = 1

21. Secure-dominating set - Main result
Theorem 17.
γs(Pn1  Pn2  …  Pnk) =

22. Secure-dominating set - Main result
P2 × P4 × P6 = P2 × G, G = P4 × P6, γs(P2 × P4 × P6) = 24
P3 × P4 × P5 = P4 × G, G = P3 × P5, γs(P3 × P4 × P5) = 30

23. Secure-dominating set - Main result
Pn1  Pn2  …  Pnk = (Pn1  P n2  …  Pnk1)  Pnk
If nk = 4l+1,
If nk = 4l+3, …
Pn1  P n2  …  Pnk1
|S*(G)| = n1n2…nk/2 …
Pn1  P n2  …  Pnk1

24. Secure-dominating set - Main result
P3 × P5 × P7 × P9 × P11 × P13
= (P3 × P5 × P7 × P9 × P11 )× P13
P3 × P5 × P7 × P9 × P11
P3 × P5 × P7 × P9
P3 × P5
P3 × P5 × P7
|S*(G)| = (3 × 5 × 7 × 9 × 11 × 13)/2

25. Secure-dominating set - Main result
Lemma 18.
In Pn1  Pn2  …  Pnk, S* is selected as previous rules, for any black super node R, there are at most four red super node Ri, 1  i  4, with wA(Ri) = 0, adjacet to R. If for all x  R − S*, x  A(u), for some u  Ri. There exists a defense D of S* corresponding to A.
S*(P5  P5  Pn) 

26. Secure-dominating set - Main result
Proof：
Pn1  Pn2  …  Pnk when n1, n2, …, nk are odd.
Case 1
nk = 4l + 3，nk1 = 4m + 3
Case 2
nk = 4l + 3，nk1 = 4m + 1
Case 3
nk = 4l + 1，nk1 = 4m + 3
Case 4
nk = 4l + 1，nk1 = 4m + 1

27. Secure-dominating set - Main result
Proof：
S*(Pn1  Pn2) is secure
If S*(Pn1  Pn2  …  Pnk-1) is secure
then S*(Pn1  Pn2  …  Pnk)：

28. Secure-dominating set - Main result
Proof：
Case 1：If nk = 4l+3, nk–1 = 4m+3 … … … … … … … ……

29. Secure-dominating set - Main result
Proof：
Case 1：

30. Secure-dominating set - Main result
|S*(G)| = (n1  n2  …  nk)/2
Theorem 17.
γs(Pn1  Pn2  …  Pnk) =

31. Secure-dominating set - Main result
Theorem 19. [2] K7 K4

32. Secure-dominating set - Main result
Theorem 20.
γs(Km1  K m2  … Kmk1  Kmk) =

33. Secure-dominating set - Main result
K2  K4  K6 = K2  (K4  K6)
K3  K4  K5  K6 = K4  (K3  K5  K6)
K3  K5  K6
K3  K4  K5  K6 K6 K6 K6 K6 K6 K6 K6 K6
K2  K4  K6

34. Secure-dominating set - Main result
Km1  K m2  … Kmk1  Kmk = (Km1  K m2  … Kmk1)  Kmk
Km1  K m2  … Kmk1 = (Km1  K m2  … Kmk2)  Kmk1 …
Km1  …  Kmk …
K m1  …  Kmk-1 … …
Km1

35. Secure-dominating set - Main result
K3  K5  K7 = (K3  K5)  K7
K3  K5  K7
K3  K5 K3
|S*(G)| = (3 × 5 × 7)/2

36. Secure-dominating set - Main result
Proof：
S*(Km1) is secure
If S*(Km1  K m2  …  Kmk1) is secure
then S*(Km1  K m2  …  Kmk)：
Kmk ok
Km1  K m2  …  Kmk1 … … …
Km1  K m2  …  Kmk1
Km1  K m2  …  Kmk1

37. Secure-dominating set Secure-dominating set
Conclusions
The Results of Previous Scholar
General graph G
G = Pn
G = Pm × Pn
G = Km
Secure set
 |V(G)| 2
min{m, n, 3}
Secure-dominating set 
Main Results
G = Pn1  Pn2  …  Pnk
G = Km1  Km2  … Kmk
Secure set
When n1 = n2 = 2,
s(G)  4n3  …  nk-2
When 2  n2,
s(G)  3n1  …  nk-2
When mk-1 is even,
s(G) 
When mk-1 is odd,
Secure-dominating set