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Secure and Secure-dominating Set of Cartesian Product Graphs

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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  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 • Secure set • Secure-dominating set • Secure set • Preliminary • Main result • Secure-dominating set • Preliminary • Main result • Conclusions 2

  3. 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.

  4. 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

  5. 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}.

  6. 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}.

  7. 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.

  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×P2P5×P5 s(G) = 2 s(G) =3

  10. 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

  11. 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 … … …

  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. Whenn1 = n2 =2, s(Pn1 P n2 … Pnk)  4n3  …  nk2 2. When2 < 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 a b l Km2 c m2 − l 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. P3× P7 P7 × P7(P3P4) × 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}. 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

  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, …, 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

  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 K6 K6 K6 K6 K3 K4 K5 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. Conclusions The Results of Previous Scholar Main Results

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