1 secure and secure-dominating set of cartesian product graphs kai-ping huang and justie su-tzu juan...

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

22

Outline

• Introduction– Secure set– Secure-dominating set

• Secure set– Preliminary– Main result

• Secure-dominating set– Preliminary– Main result

• Conclusions

3

N(v)

Introduction

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) =∪vSN(v).4. N[S]= N(S) ∪ S.

N[S]

S

v

4

Introduction

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.

G Su

v1

2

3

A(u) ={1, 2}A(v) ={3}D(u) ={u}D(v) ={v}

D(u) = {u, v}D(v) = ∅

4

A(u) = {2}A(v) = {1, 3}

Gu

v1

2

3

5

Introduction

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}.

S

6

IntroductionG S

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

Introduction

General graph G

Pn Pm × Pn Km

Pn1 Pn2

… Pnk

Km1 Km2

… Kmk

Secure set Brigham et al, 2007 [1]

The thesisSecure-dominating

setChang et al, 2008 [2]

[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.

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

Secure set - Main result

• Theorem 4.

1 < n1 n2 … nk1 nk

1. When n1 = n2 =2,

s(Pn1 P n2

… Pnk) 4n3 … nk2

2. When 2 < n2,

s(Pn1 P n2

… Pnk) 3n1 n2 … nk2

10

11


• 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

…

… …

12


• 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


• 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 … Pnk2

) Pnk1 Pnk

1. When n1 = n2 =2,

s(Pn1 P n2

… Pnk) 4n3 … nk2

2. When 2 < n2,

s(Pn1 P n2

… Pnk) 3n1 n2 … nk2

14


• Theorem 6. [1]

s(Km) =

2

m

K4

K7

15


• Theorem 7.

1.When mk1 is even,

2.When mk1 is odd,

• Km1 K m2

… Kmk1 Kmk

= (Km1 K m2

… Kmk1) Kmk

2

...... 21

21

kmmm

mmmKKKs

k

k

k

kkmmm m

m

mmmmKKKs

k 32

1......

1

21

22121

16


• Km1 K m2

if m1 odd

even

Km1

Km2

21m

21m

l

m2 − l

211

1

222

2m

mm

m

l

lmm

lm

lm

2111

222

21

1

3

1m

m

ml

2ml

a b

c

17


• 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,

2

2121

mmKKs mm

2

1

21

32

121

mm

mKKs mm

2

...... 21

21

kmmm

mmmKKKs

k

k

k

kkmmm m

m

mmmmKKKs

k 32

1......

1

21

22121

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 ：

① γs(G × Pn) = n/2 |V(G)|.

② V(Pn) = {v1,v2, ··· ,vn} , S = {(u, vi): u ∈ V(G), i ≡ 2, 3 (mod 4)} is a secure-dominating set.

P5 × P8

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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

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Secure-dominating set - Preliminary

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.

)1(1 Aw

wA(v3) = 1

wA(v2) = 0wA(v1) = 1

21

Secure-dominating set - Main result

• Theorem 17.

γs(Pn1 Pn2

… Pnk) =

2

...21 knnn

22


• 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

• Pn1 Pn2

… Pnk = (Pn1

P n2 … Pnk1

) Pnk

If nk = 4l+1,

If nk = 4l+3,


23

Pn1 P n2

… Pnk1

Pn1 P n2

… Pnk1

…

…

|S*(G)| = n1n2…nk/2


P3 × P5 × P7 × P9 × P11 × P13

= (P3 × P5 × P7 × P9 × P11 )× P13

24

P3 × P5 × P7 × P9 × P11

P3 × P5 × P7 × P9

P3 × P5 × P7

P3 × P5

|S*(G)| = (3 × 5 × 7 × 9 × 11 × 13)/2

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• 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)

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• Proof ：Pn1

Pn2 … Pnk

when n1, n2, …, nk are 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

27


• Proof ：S*(Pn1

Pn2) is secure

If S*(Pn1 Pn2

… Pnk-1) is secure

then S*(Pn1 Pn2

… Pnk) ：

…

• Proof ：Case 1 ： If nk = 4l+3, nk–1 = 4m+3


……………

……

…

• Proof ：Case 1 ：


30


|S*(G)| = (n1 n2 … nk)/2

• Theorem 17.

γs(Pn1 Pn2

… Pnk) =

2

...21 knnn

31


• Theorem 19. [2]

2

mKsK mm

s

K4

K7

32


• Theorem 20.

γs(Km1 K m2

… Kmk1 Kmk

) =

2

...21 kmmm

33


• K2 K4 K6 = K2 (K4 K6)

• K3 K4 K5 K6 = K4 (K3 K5 K6)

K3 K5 K6

K3 K4 K5 K6

K6

K6K6

K6K6

K6K6

K6

K2 K4 K6

34


• Km1 K m2

… Kmk1 Kmk

= (Km1 K m2

… Kmk1) Kmk

Km1 K m2

… Kmk1 = (Km1

K m2 … Kmk2

) Kmk1

……

Km1 … Kmk

2

1km

…

K m1 … Kmk-1

2

11 km

…

Km1

2

1m

35


• K3 K5 K7 = (K3 K5) K7

K3 K5 K7 K3 K5 K3

|S*(G)| = (3 × 5 × 7)/2


• Proof ：S*(Km1

) is secure

If S*(Km1 K m2

… Kmk1) is secure

then S*(Km1 K m2

… Kmk) ：

36

…

Kmk ok

… …

Km1 K m2

… Kmk1

Km1 K m2

… Kmk1

Km1 K m2

… Kmk1

37

Conclusions

General graph G G = Pn G = Pm × Pn G = Km

Secure set |V(G)| 2 min{m, n, 3}

Secure-dominating set

2

)(GV

2

)(GV

2

m

G = Pn1 Pn2

… PnkG = 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,s(G)

Secure-dominating set

2

)(GV

2

)(GV

3

1

2

)(

11

21

kk

k

mm

mGV

The Results of Previous Scholar

Main Results

1 secure and secure-dominating set of cartesian product graphs kai-ping huang and justie su-tzu juan...

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