The 3 * -connected property of the pyramid networks

1. The 3*-connected property of the pyramid networks Yuan-Hsiang Teng, Tzu-Liang Kung, Lih-Hsing Hsu Computers and Mathematics with Applications 60 (2010) 2360–2363

2. Abstract • We prove that the pyramid network PM[n] is 3*-connected and super connected for n ≥ 1.

3. Definition and Notation for a Graph • G=(V,E) is a graph if V is a finite set and E is a subset of {(a,b) | (a,b) is an unordered pair of V}. • V is the node set and E is the edge set of G.

4. Hamiltonian Properties • A hamiltonian path is a path such that its nodes are distinct and span V. • A hamiltonian cycle is a cycle such that its nodes are distinct except for the first node and the last node and span V. • A hamiltonian graph is a graph with a hamiltonian cycle.

5. Connectivity • The connectivity of G, κ(G) is the minimum number of vertices whose removal leaves the remaining graph disconnected or trivial.

6. Container • A k-container C(u,v) in a graph G is a set of k disjoint paths joining u to v. • A k*-container C(u,v) in a graph G is a k-container such that every vertex of G is on some path in C(u,v).

7. k*-connected • A graph G is k*-connected if there exists a k*-container between any two distinct vertices in G. • A graph G is 1*-connected if and only if it is hamiltonian connected. • A graph G is 2*-connected if it is hamiltonian. • The study of k*-connected graph is motivated by the 3*-connected graphs proposed by Albert et al. • M. Albert, E.R.L. Aldred, D. Holton, J. Sheehan, On 3*-connected graphs, Australasian Journal of Combinatorics 24 (2001) 193-207.

8. Super connected • A graph G is super connected if it is i*-connected for all 1≤i ≤κ(G). • H.C. Hsu, C.K. Lin, H.M. Huang, L.H. Hsu, The spanning connectivity of the (n, k)-star graphs, International Journal of Foundations of Computer Science • 17 (2006) 415–434. • C.K. Lin, H.M. Huang, L.H. Hsu, The super connectivity of the pancake graphs and the super laceability of the star graphs, Theoretical Computer Science • 339 (2005) 257–271.

9. Introduction • κ(PM[n])=3 • Sarbazi-Azad et al. andWu et al. studied the hamiltonicity and the hamiltonian connectivityof the pyramid networks. • Thus the pyramid network is 1*-connected and 2*-connected. • H. Sarbazi-Azad, H.M. Ould-Khaoua, L.M. Mackenzie, Algorithmic construction ofhamiltonians in pyramids, Information Processing Letters 80 (2001) 75-79. • R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault, Journal fo Information Science and Engineering 25 (2009) 531-542.

10. Introduction • In this paper, we study the 3*-connected property of the pyramid network. • We prove that any pyramid network is 3*-connected. • Moreover, the pyramid network is super connected.

11. The pyramid networks • An n-dimensional pyramid network PM[n] is a hierarchy structure based on mesh networks. • The subgraph induced by all vertices in the i-th layer of a PM[n] is a mesh network M(2i, 2i). • V (PM[n]) = {(k; x; y) | 0 ≤k ≤ n; 1 ≤ x ≤2k; 1 ≤y ≤2k}.

12. The pyramid networks • Two vertices (k1; x1; y1) and (k2; x2; y2) in PM[n] are adjacent if they satisfy one of the following conditions:

13. The pyramid networks PM[2]

14. Lemma 1 • A pyramid network PM[n] is hamiltonian for n ≥1. • H. Sarbazi-Azad, H.M. Ould-Khaoua, L.M. Mackenzie, Algorithmic construction of Hamiltonians in pyramids, Information Processing Letters 80 (2001) • 75–79. • R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault, Journal of Information Science and Engineering 25 (2009) 531–542.

15. Lemma 2 • A pyramid network PM[n] is hamiltonian connected for n ≥1. R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault, Journal of Information Science and Engineering 25 (2009) 531–542.

16. Lemma 3 • A pyramid network PM[n] with one vertex fault is hamiltonian for n ≥1. R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault, Journal of Information Science and Engineering 25 (2009) 531–542.

17. Lemma 4 • A mesh network M(m,n) is hamiltonian laceable except : (1) m = 2 and n ≠ 2, and (2) m = 3 and n = 2m. G. Simmons, Almost all n-dimensional rectangular lattices are Hamiltonian laceable, Congressus Numerantium (1978) 103–108.

18. Lemma 5 • A mesh network M(m,n) is hamiltonian for m = n.

19. Theorem 1 • Assume that n ≥1. • Let s and t be any two distinct vertices of a pyramid network PM[n]. • Then there exists a 3*-container C3*(s,t) of PM[n].

20. Proof of Theorem 1 • We prove this theorem by induction. • By brute force, we check the theorem holds for n = 1. • Assume the theorem holds for any PM[n-1] with n > 1.

21. Case 1: s = (k1; x, y) and t = (k2;w, z) with 1 ≤ k1, k2≤n -1. hypothesis s t PM[n-1] (n-1)-layer Lemma 4 n-th layer

22. Case 2.1: s = (n; x, y) and t = (n;w, z) Lemma 2 PM[n-1] (n-1)-layer s t n-th layer Lemma 5

23. Case 2.2: s = (n; x, y) and t = (n;w, z) Lemma 3 PM[n-1] u (n-1)-layer s t n-th layer

24. Case 3.1~Case 3.4: s = (k; x, y) and t = (n;w, z) with 1≤ k≤n-1 exceptt∊{(n;1,1); (n; 1, 2n); (n; 2n, 1); (n;2n, 2n)}. hypothesis s PM[n-1] (n-1)-layer t’ Algorithm n-th layer t

25. Case 4: s = (k; x, y) and t ∊{(n;1,1); (n; 1, 2n); (n; 2n, 1); (n;2n, 2n)}. Case 4.1: s ≠ t’ hypothesis s PM[n-1] (n-1)-layer t’ Algorithm n-th layer t

26. Case 4: s = (k; x, y) and t ∊{(n;1,1); (n; 1, 2n); (n; 2n, 1); (n;2n, 2n)}. Case 4.2: s = t’ PM[n-1] (n-1)-layer s Lemma 3 n-th layer t

27. Corollary 1 • The pyramid network PM[n] is super connected for n ≥1. • By Lemma 1, Lemma 2, and Theorem 1, the pyramid network is k*-connected for 1 ≤k ≤3.