Loading in 5 sec....

Secure and Secure-dominating Set of Cartesian Product GraphsPowerPoint Presentation

Secure and Secure-dominating Set of Cartesian Product Graphs

- 165 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Secure and Secure-dominating Set of Cartesian Product Graphs' - rupert

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### 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) =∪vSN(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 uS and A(u) ∩ A(v) = ∅

for all u, vS, u v.

6. D : S → Ƥ (S) is called a defense of S if D(u) ⊆ N[u] ∩

S for all uS and D(u) ∩ D(v) = ∅ for all u, vS,

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

also adominating set ofG.

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.

- Corollary 2.[1]

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 … nk1 nk

1. Whenn1 = n2 =2,

s(Pn1 P n2 … Pnk) 4n3 … nk2

2. When2 < n2,

s(Pn1 P n2 … Pnk) 3n1 n2 … nk2

Secure set - Main result

- s(Pn1×Pn2×Pn3), n1n2n3
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), n1n2 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(Pn1Pn2Pn3) 4

2. When 2 < n2, s(Pn1Pn2Pn3) 3n1

Pn1 Pn2 … Pnk = (Pn1 Pn2 … Pnk2 ) Pnk1 Pnk

1. Whenn1 = n2 =2,

s(Pn1 P n2 … Pnk) 4n3 … nk2

2. When2 < n2,

s(Pn1 P n2 … Pnk) 3n1 n2 … nk2

Secure set - Main result

- Theorem 7.
1.When mk1 is even,

2.When mk1 is odd,

- Km1 K m2 … Kmk1 Kmk = (Km1 K m2 … Kmk1) Kmk

Secure set - Main result

- Lemma 8.
1.When m1 is even,

2.When m1 is odd,

Km1 K m2 … Kmk1 Kmk = (Km1 K m2 … Kmk1) Kmk

1.When mk1 is even,

2.When mk1 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(P3P4) × 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 … Pnk1) Pnk
If nk= 4l+1,

If nk= 4l+3,

…

Pn1 P n2 … Pnk1

|S*(G)| = n1n2…nk/2

…

Pn1 P n2 … Pnk1

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，nk1= 4m + 3

- Case 2
nk= 4l + 3，nk1= 4m + 1

- Case 3
nk= 4l + 1，nk1= 4m + 3

- Case 4
nk= 4l + 1，nk1= 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：

Secure-dominating set- Main result

- Theorem 20.
γs(Km1 K m2 … Kmk1 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 … Kmk1 Kmk = (Km1 K m2 … Kmk1) Kmk
Km1 K m2 … Kmk1=(Km1 K m2 … Kmk2) Kmk1

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 … Kmk1) is secure

then S*(Km1 K m2 … Kmk)：

Kmk

ok

Km1 K m2 … Kmk1

…

…

…

Km1 K m2 … Kmk1

Km1 K m2 … Kmk1

Download Presentation

Connecting to Server..