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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.

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Presentation on theme: "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."— Presentation transcript:

1 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 22 Outline Introduction –Secure set –Secure-dominating set Secure set –Preliminary –Main result Secure-dominating set –Preliminary –Main result Conclusions

3 3 N(v)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) = ∪ v  S N(v). 4. N[S]= N(S) ∪ S. N[S]N[S] S v

4 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 S u v 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}

5 G u v 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 6 Introduction G 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. γ s (G) = min{|S| : S is a secure-dominating set of G}.

7 7 Introduction General graph G PnPn P m × P n KmKm P n 1  P n 2  …  P n k K m 1  K m 2  …  K m k Secure setBrigham 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 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 S 1 and S 2 are vertex disjoint secure sets in the same graph, then S 1 ∪ S 2 is a secure set.

9 9 Secure set - Preliminary Proposition 3. [1] s(P m × P n ) = min{m, n, 3}. P 3 × P 2 P 5 × P 5 s(G) = 2 s(G) =3

10 Secure set - Main result Theorem 4. 1 < n 1  n 2  …  n k  1  n k 1. When n 1 = n 2 =2, s(P n 1  P n 2  …  P n k )  4n 3  …  n k  2 2. When 2 < n 2, s(P n 1  P n 2  …  P n k )  3n 1  n 2  …  n k  2 10

11 11 Secure set - Main result s(P n 1 × P n 2 × P n 3 ), n 1  n 2  n 3 P 2 × P 2 × P 2 P 2 × P 3 × P 3 P 3 × P 3 × P 3 P 2 × P 2 × P 3 P 2 × P 3 × P 4 P 3 × P 3 × P 4 … … …

12 12 Secure set - Main result s(P n 1 × P n 2 × P n 3 ), n 1  n 2  n 3 G = P 4 × P n 2 × P n 3, s(G)  12 G = P 5 × P n 2 × P n 3, s(G)  15 G = P n 1 × P n 2 × P n 3, s(G)  3n 1 …

13 13 Secure set - Main result Lemma When n 1 = n 2 =2, s(P n 1  P n 2  P n 3 )  4 2. When 2 < n 2, s(P n 1  P n 2  P n 3 )  3n 1 P n 1  P n 2  …  P n k = (P n 1  P n 2  …  P n k  2 )  P n k  1  P n k 1. When n 1 = n 2 =2, s(P n 1  P n 2  …  P n k )  4n 3  …  n k  2 2. When 2 < n 2, s(P n 1  P n 2  …  P n k )  3n 1  n 2  …  n k  2

14 14 Secure set - Main result Theorem 6. [1] s(K m ) = K4K4 K7K7

15 15 Secure set - Main result Theorem 7. 1.When m k  1 is even, 2.When m k  1 is odd, K m 1  K m 2  …  K m k  1  K m k = (K m 1  K m 2  …  K m k  1 )  K m k

16 16 Secure set - Main result K m 1  K m 2 if m 1 odd even Km1Km1 Km2Km2 l m 2 − l ab c

17 17 Secure set - Main result Lemma 8. 1.When m 1 is even, 2.When m 1 is odd, K m 1  K m 2  …  K m k  1  K m k = (K m 1  K m 2  …  K m k  1 )  K m k 1.When m k  1 is even, 2.When m k  1 is odd,

18 18 Secure-dominating set - Preliminary Theorem 9. [2] For any graph G with |V(G)| = n, γ s (G) ≥  n/2 . Theorem 10. [2] For all n ≥ 2, γ s (P n ) =  n/2 . Corollary 11. [2] 1. γ s (G × P n ) ≤  n/2  |V(G)|. 2. When n is even : γ s (G × P n ) =  n/2  |V(G)|. ‚ V(P n ) = {v 1,v 2, ···,v n }, S = {(u, v i ): u ∈ V(G), i ≡ 2, 3 (mod 4)} is a secure-dominating set. P 5 × P 8

19 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 P 3 × P n. 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 P 5 × P n. Theorem 14. [2] For all m, n ≥ 2, γ s (P m × P n ) =  mn/2 . P 3 × P 7 P 7 × P 7  (P 3  P 4 ) × P 7

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

21 Secure-dominating set - Main result Theorem 17. γ s (P n 1  P n 2  …  P n k ) = 21

22 22 Secure-dominating set - Main result P 2 × P 4 × P 6 = P 2 × G, G = P 4 × P 6, γ s (P 2 × P 4 × P 6 ) = 24 P 3 × P 4 × P 5 = P 4 × G, G = P 3 × P 5, γ s (P 3 × P 4 × P 5 ) = 30

23 P n 1  P n 2  …  P n k = (P n 1  P n 2  …  P n k  1 )  P n k If n k = 4l+1, If n k = 4l+3, Secure-dominating set - Main result 23 P n 1  P n 2  …  P n k  1 … … |S * (G)| =  n 1 n 2 …n k /2 

24 Secure-dominating set - Main result P 3 × P 5 × P 7 × P 9 × P 11 × P 13 = (P 3 × P 5 × P 7 × P 9 × P 11 )× P P 3 × P 5 × P 7 × P 9 × P 11 P 3 × P 5 × P 7 × P 9 P 3 × P 5 × P 7 P3 × P5P3 × P5 |S * (G)| =  (3 × 5 × 7 × 9 × 11 × 13)/2 

25 Lemma 18. In P n 1  P n 2  …  P n k, S * is selected as previous rules, for any black super node R, there are at most four red super node R i, 1  i  4, with w A (R i ) = 0, adjacet to R. If for all x  R − S *, x  A(u), for some u  R i. There exists a defense D of S * corresponding to A. Secure-dominating set - Main result 25 S * (P 5  P 5  P n ) 

26 Secure-dominating set - Main result Proof : P n 1  P n 2  …  P n k when n 1, n 2, …, n k are odd. Case 1 n k = 4l + 3 , n k  1 = 4m + 3 Case 2 n k = 4l + 3 , n k  1 = 4m + 1 Case 3 n k = 4l + 1 , n k  1 = 4m + 3 Case 4 n k = 4l + 1 , n k  1 = 4m

27 Secure-dominating set - Main result Proof : S*(P n 1  P n 2 ) is secure If S*(P n 1  P n 2  …  P n k-1 ) is secure then S*(P n 1  P n 2  …  P n k ) : 27

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

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

30 Secure-dominating set - Main result |S * (G)| =  (n 1  n 2  …  n k )/2  Theorem 17. γ s (P n 1  P n 2  …  P n k ) = 30

31 31 Secure-dominating set - Main result Theorem 19. [2] K4K4 K7K7

32 32 Secure-dominating set - Main result Theorem 20. γ s (K m 1  K m 2  …  K m k  1  K m k ) =

33 33 Secure-dominating set - Main result K 2  K 4  K 6 = K 2  (K 4  K 6 ) K 3  K 4  K 5  K 6 = K 4  (K 3  K 5  K 6 ) K 3  K 5  K 6 K 3  K 4  K 5  K 6 K6K6 K6K6 K6K6 K6K6 K6K6 K6K6 K6K6 K6K6 K 2  K 4  K 6

34 34 Secure-dominating set - Main result K m 1  K m 2  …  K m k  1  K m k = (K m 1  K m 2  …  K m k  1 )  K m k K m 1  K m 2  …  K m k  1 = (K m 1  K m 2  …  K m k  2 )  K m k  1 … … K m 1  …  K m k … K m 1  …  K m k-1 … Km1Km1

35 35 Secure-dominating set - Main result K 3  K 5  K 7 = (K 3  K 5 )  K 7 K 3  K 5  K 7 K 3  K 5 K3K3 |S * (G)| =  (3 × 5 × 7)/2 

36 Secure-dominating set - Main result Proof : S * (K m 1 ) is secure If S * (K m 1  K m 2  …  K m k  1 ) is secure then S * (K m 1  K m 2  …  K m k ) : 36 … KmkKmk ok … … K m 1  K m 2  …  K m k  1

37 37 Conclusions General graph GG = P n G = P m × P n G = K m Secure set  |V(G)| 2min{m, n, 3} Secure-dominating set  G = P n 1  P n 2  …  P n k G = K m 1  K m 2  …  K m k Secure set When n 1 = n 2 = 2, s(G)  4n 3  …  n k-2 When 2  n 2, s(G)  3n 1  …  n k-2 When m k-1 is even, s(G)  When m k-1 is odd, s(G)  Secure-dominating set The Results of Previous Scholar Main Results


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