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

Introduction Def: Let G = (V, E) be a graph. If v  V and S ⊆ V :
N[S] Introduction N(v) v S 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SN(v). 4. N[S]= N(S) ∪ S.

A(u) ={1, 2} A(v) ={3} A(u) = {2} A(v) = {1, 3} Introduction G S 2 u D(u) ={u} D(v) ={v} D(u) = {u, v} D(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. 即為在圖g上找了一個集合s，我們把跟s相鄰且不在s上的點稱為attack，則定義a(u)代表不在s集合內而跟u相鄰的攻擊函數，且規定當定義其為a(u)時則不能在定義其為a(v) 同理d為在s集合中，跟s相鄰且在s中的點稱為defense，定義d(u)為在s集合內而跟u相鄰的防禦函數，且規定當定義為d(u)時則不能在定義其為d(v) 4

Introduction S G u v 1 2 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}. 對所有在s上的攻擊a而言，都能找到對應攻擊a的有效防禦

Introduction 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}.

Secure-dominating set
Introduction General graph G Pn Pm × Pn Km Pn1  Pn2  …  Pnk Km1  Km2  … Kmk Secure set Brigham 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.

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 × P2 P5 × P5 s(G) = 2 s(G) =3

Secure set - Main result
Theorem 4. 1 < n1  n2  …  nk1  nk 1. When n1 = n2 =2, s(Pn1  P n2  …  Pnk)  4n3  …  nk2 2. When 2 < 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 × P P3 × P3 × P3 P2 × P2 × P3 P2 × P3 × P 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. When n1 = n2 =2, s(Pn1  P n2  …  Pnk)  4n3  …  nk2 2. When 2 < 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

Secure set - Main result
Km1  K m2 if m1 odd even l a b Km2 m2 − l c 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. P7 × P7 (P3  P4) × P7 P3 × 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}. = 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, …, nk are 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 ……

Secure-dominating set - Main result
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 K3  K4  K5  K6 K6 K6 K6 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

Secure-dominating set Secure-dominating set
Conclusions The Results of Previous Scholar General graph G G = Pn G = Pm × Pn G = Km Secure set  |V(G)| 2 min{m, n, 3} Secure-dominating set Main Results G = Pn1  Pn2  …  Pnk G = 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, Secure-dominating set