Y-Chuang Chen

Restricted connectivity for three families of interconnection networks

Vertex connectivity and edge connectivity are two important parameters in interconnection networks. Even though they reflect the fault tolerance correctly, they undervalue the resilience of large networks. By the concept of conditional connectivity and super-connectivity, the concept of restricted vertex connectivity and restricted edge connectivity of graphs was proposed by Esfahanian [A.H. Esfahanian, Generalized measures of fault tolerance with application to N-cube networks, IEEE Transactions on Computers 38 (1989) 1586–1591]. Such measures take the resilience of large networks into consideration. In this paper, we propose three families of interconnection networks and discuss their restricted vertex connectivity and restricted edge connectivity. In particular, the hypercubes, twisted-cubes, crossed-cubes, mobius cubes, star graphs, pancake graphs, recursive circulant graphs, and k-ary n-cubes are special cases of these families.

On some super fault-tolerant Hamiltonian graphs

A k-regular Hamiltonian and Hamiltonian connected graph G is super fault-tolerant Hamiltonian if G remains Hamiltonian after removing at most k-2 nodes and/or edges and remains Hamiltonian connected after removing at most k-3 nodes and/or edges. A super fault-tolerant Hamiltonian graph has a certain optimal flavor with respect to the fault-tolerant Hamiltonicity and Hamiltonian connectivity. In this paper, we investigate a construction scheme to construct super fault-tolerant Hamiltonian graphs. In particularly, twisted-cubes, crossed-cubes, and Mobius cubes are all special cases of this construction scheme. Therefore, they are all super fault-tolerant Hamiltonian graphs.

Topological Properties on the Wide and Fault Diameters of Exchanged Hypercubes

The n-dimensional hypercube is one of the most popular topological structure for interconnection networks in parallel computing and communication systems. The exchanged hypercube EH(s, t), a variant of the hypercube, retains several valuable and desirable properties of the hypercube such as a small diameter, bipancyclicity, and super connectivity. In this paper, we construct s + 1 (or t + 1) internally vertex-disjoint paths between any two vertices for parallel routes in the exchanged hypercube EH(s, t) for 3 ≤ s ≤ t. We also show that both the (s + 1)-wide diameter and s-fault diameter of the exchanged hypercube EH(s, t) are s + t + 3 for 3 ≤ s ≤ t.

Optimal Edge Congestion of Exchanged Hypercubes

Topological properties have become a popular and important area of focus for studies that analyze interconnections between networks. The hypercube is one of the most widely discussed topological structures for interconnections between networks and is usually covered in introductions to the basic principles and methods for network design. The exchanged hypercube EH(s, t) is a new variant of the hypercube that has slightly more than half as many edges and retains several valuable and desirable properties of the hypercube. In this paper, we propose an approach for shortest path routing algorithms from the source vertex to the destination vertex in EH(s, t) with time complexity O(n), where n = s + t + 1 and 1 ≤ s ≤ t. We focus on edge congestion, which is an important indicator for cost analyses and performance measurements in interconnection networks. Based on our shortest path routing algorithm, we show that the edge congestion of EH(s, t) is 3 · 2s+t+1 - 2s+1 - 2t+1. In addition, we prove that our shortest path routing algorithm is an optimal routing strategy with respect to the edge congestion of EH(s, t).

Maximally Local Connectivity on Augmented Cubes

Connectivity is an important measurement for the fault tolerance in interconnection networks. It is known that the augmented cube AQ n is maximally connected , i.e. (2n - 1)-connected, for n *** 4. By the classical Mengers Theorem , every pair of vertices in AQ n is connected by 2n - 1 vertex-disjoint paths for n *** 4. A routing with parallel paths can speed up transfers of large amounts of data and increase fault tolerance. Motivated by some research works on networks with faults, we have a further result that for any faulty vertex set F *** V (AQ n ) and |F | ≤ 2n *** 7 for n *** 4, each pair of non-faulty vertices, denoted by u and v , in AQ n *** F is connected by min{deg f (u ), deg f (v )} vertex-disjoint fault-free paths, where deg f (u ) and deg f (v ) are the degree of u and v in AQ n *** F , respectively. Moreover, we have another result that for any faulty vertex set F *** V (AQ n ) and |F | ≤ 4n *** 9 for n *** 4, there exists a large connected component with at least 2 n *** |F | *** 1 vertices in AQ n *** F . In general, a remaining large fault-free connected component also increases fault tolerance.

A family of Hamiltonian and Hamiltonian connected graphs with fault tolerance

Processor (vertex) faults and link (edge) faults may happen when a network is used, and it is meaningful to consider networks (graphs) with faulty processors and/or links. A k-regular Hamiltonian and Hamiltonian connected graph G is optimal fault-tolerant Hamiltonian and Hamiltonian connected if G remains Hamiltonian after removing at most k−2 vertices and/or edges and remains Hamiltonian connected after removing at most k−3 vertices and/or edges. In this paper, we investigate in constructing optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected graphs. Therefore, some of the generalized hypercubes, twisted-cubes, crossed-cubes, and Möbius cubes are optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected.

A Note on Diagnosability of Large Fault Sets on Star Graphs

Diagnosability of a system directly refers to the maximum number of faulty vertices that can be identified by the system. Somani et al. [2] proposed a generalized measure to increase the degree of diagnosability of the hypercubes and star graphs. This paper provides counterexamples for the results of diagnosability of star graphs.

Maximally local connectivity and connected components of augmented cubes

Abstract The connectivity of a graph is an important issue in graph theory, and is also one of the most important factors in evaluating the reliability and fault tolerance of a network. It is known that the augmented cube AQ n is maximally connected, i.e. ( 2 n - 1 ) -connected, for n ⩾ 4 . By the classic Menger’s Theorem, every pair of vertices in AQ n is connected by 2 n - 1 vertex-disjoint paths for n ⩾ 4 . A routing with parallel paths can speed up transfers of large amounts of data and increase fault tolerance. Motivated by research on networks with faults, we obtained the result that for any faulty vertex set F ⊂ V ( AQ n ) and | F | ⩽ 2 n - 7 for n ⩾ 4 , each pair of non-faulty vertices, denoted by u and v, in AQ n - F is connected by min { deg f ( u ) , deg f ( v ) } vertex-disjoint fault-free paths, where deg f ( u ) and deg f ( v ) are the degree of u and v in AQ n - F , respectively. Moreover, we demonstrate that for any faulty vertex set F ⊂ V ( AQ n ) and | F | ⩽ 4 n - 9 for n ⩾ 4 , there exists a large connected component with at least 2 n - | F | - 1 vertices in AQ n - F , which improves on the results of Ma et al. (2008) who show this for n ⩾ 6 .

Internally Disjoint Paths in a Variant of the Hypercube

The hypercube is one of the most popular interconnection networks for parallel computer/communication system. The exchanged hypercube, which is a variant of the hypercube, maintains several desirable properties of the hypercube such as low diameter, bipancyclicity, and super connectivity. In this paper, we give internally disjoint paths for parallel routing in exchanged hypercubes and show the wide diameter of exchanged hypercubes.

Hamiltonian decomposition of generalized recursive circulant graphs

GRC graphs have more flexible structures than recursive circulant graphs.We construct edge-disjoint Hamiltonian cycles of GRC graphs.We prove that some of the GRC graphs are Hamiltonian decomposable. In 2012, Tang et al. 9 proposed a new class of graphs called generalized recursive circulant (GRC) graphs, which is an extension of recursive circulant graphs. GRC graphs have a more flexible structure than recursive circulant graphs, while retaining their attractive properties, such as degree, connectivity, diameter, and routing algorithm. In this paper, the Hamiltonian decomposition of some GRC graphs is discussed.