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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. Outline. Introduction S ecure set S ecure-dominating set S ecure set Preliminary Main result

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Secure and Secure-dominating Set of Cartesian Product Graphs

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## 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

### Outline

• Introduction

• Secure set

• Secure-dominating set

• Secure set

• Preliminary

• Main result

• Secure-dominating set

• Preliminary

• Main result

• Conclusions

2

### Introduction

N[S]

v

N(v)

S

Def:

Let G = (V,E) be a graph. Ifv 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.

### Introduction

A(u) = {2}

A(v) = {1, 3}

A(u) ={1, 2}

A(v) ={3}

G

S

u

2

D(u) = {u, v}

D(v) = ∅

D(u) ={u}

D(v) ={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.

4

### Introduction

S

G

u

2

1

v

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}.

### Introduction

G

S

Def:

9. Dominating set:G if N[S] = V(G).

10. Secure-dominatingset : S is a secure set of G that is

11. γs(G)=min{|S| : S is a secure-dominating set of G}.

### Introduction

[1] R. C. Brigham, R. D. Dutton, S. T. Hedetniemi, “Security in graphs,” Discrete Appl. Math., 155 (2007), 1708-1714.

[2] Chia-Lang Chang, Tsui-Ping Chang, David Kuo, “Secure and secure-dominating set of graphs,” National Dong Hwa University Applied Mathematics, Manuscript.

### 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.

### Secure set - Preliminary

• Proposition 3. [1]

s(Pm × Pn) = min{m, n, 3}.

P3×P2P5×P5

s(G) = 2 s(G) =3

### Secure set - Main result

• Theorem 4.

1 < n1  n2  …  nk1  nk

1. Whenn1 = n2 =2,

s(Pn1 P n2 …  Pnk)  4n3  …  nk2

2. When2 < n2,

s(Pn1 P n2 …  Pnk)  3n1  n2  …  nk2

### Secure set - Main result

• s(Pn1×Pn2×Pn3), n1n2n3

P2×P2×P2 P2×P3×P3 P3×P3×P3

P2×P2×P3 P2×P3×P4 P3×P3×P4

### 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

### 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. Whenn1 = n2 =2,

s(Pn1 P n2 … Pnk)  4n3  …  nk2

2. When2 < n2,

s(Pn1 P n2 … Pnk)  3n1  n2  …  nk2

### Secure set - Main result

• Theorem 6. [1]

s(Km) =

K7

K4

### 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

• Km1 K m2

if m1 odd

even

a

b

l

Km2

c

m2 − l

Km1

### 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,

### 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

### 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.

P3× P7

P7 × P7(P3P4) × P7

### 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}.

2. = 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

### Secure-dominating set- Main result

• Theorem 17.

γs(Pn1 Pn2 … Pnk) =

### 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

### 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

### 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

### 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) 

### Secure-dominating set- Main result

• Proof：

Pn1 Pn2 …  Pnk when n1, n2, …, nkare 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

### Secure-dominating set- Main result

• Proof：

S*(Pn1 Pn2) is secure

If S*(Pn1 Pn2 …  Pnk-1) is secure

then S*(Pn1 Pn2 … Pnk)：

### Secure-dominating set- Main result

• Proof：

Case 1：If nk = 4l+3, nk–1 = 4m+3

……

• Proof：

Case 1：

### Secure-dominating set- Main result

|S*(G)| = (n1 n2 … nk)/2

• Theorem 17.

γs(Pn1 Pn2 … Pnk) =

### Secure-dominating set - Main result

• Theorem 19. [2]

K7

K4

### Secure-dominating set- Main result

• Theorem 20.

γs(Km1 K m2 … Kmk1 Kmk) =

### Secure-dominating set- Main result

• K2 K4 K6 = K2 (K4 K6)

• K3 K4 K5 K6 = K4 (K3 K5  K6)

K3 K5  K6

K6

K6

K6

K6

K3 K4 K5 K6

K6

K6

K6

K6

K2 K4 K6

### 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

### Secure-dominating set- Main result

• K3 K5 K7 = (K3 K5)  K7

K3 K5 K7

K3 K5

K3

|S*(G)| = (3 × 5 × 7)/2

### 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

### Conclusions

The Results of Previous Scholar

Main Results