Twisted hypercube-like networks (THLNs) are an important class of interconnection networks for parallel computing systems, which include most popular variants of the hypercubes, such as crossed cubes, M\"obius cubes, twisted cubes and locally twisted cubes. This paper deals with the fault-tolerant hamiltonian connectivity of THLNs under the large fault model. Let $G$ be an $n$-dimensional THLN and $F \subseteq V(G)\bigcup E(G)$, where $n \geq 7$ and $|F| \leq 2n - 10$. We prove that for any two nodes $u,v \in V(G - F)$ satisfying a simple necessary condition on neighbors of $u$ and $v$, there exists a hamiltonian or near-hamiltonian path between $u$ and $v$ in $G-F$. The result extends further the fault-tolerant graph embedding capability of THLNs.
The performance of a parallel computing system heavily depends on the effectiveness of the underlying interconnection network. An interconnection network is usually represented by a graph, where nodes and edges correspond to processors and communication links between processors, respectively. In the design and analysis of an interconnection network, one major concern is its graph embedding capability, which reflects how efficiently a parallel algorithm with structured task graph (guest graph) can be executed on this network (host graph). Cycles and paths are recognized as important guest graphs because a great number of parallel algorithms, such as matrix-vector multiplication, Gaussian elimination and bitonic sorting, have been developed on cycle/path-structured task graphs [15].
As the size of a parallel computing system increases, it becomes much likely that some processors and communication links fail to work in such a system. Consequently, it is essential to study the fault-tolerant graph embedding capability of an interconnection network with faulty elements.
The hypercube-like networks (HLNs) are an important class of generalizations of the popular hypercube interconnection networks for parallel computing. Among HLNs one may identify a subclass of networks, called the twisted hypercube-like networks (THLNs), which include most October 30, 2018 DRAFT well-known variants of the hypercubes, such as crossed cubes [5], Möbius cubes [1], twisted cubes [10] and locally twisted cubes [23]. The fault-free and fault-tolerant cycle/path embedding capabilities of these hypercube variants have been intensively studied in the literature [3], [4], [8], [9], [11], [12], [14], [19]- [22], [24].
In recent years, the fault-tolerant cycle/path embedding capabilities of HLNs and THLNs have received considerable research attention [6], [7], [13], [16]- [18], [25]. However, most of the embeddings tolerate no more faulty elements than the degree of the graph, i.e., under the small fault model. Recently, Yang et al. [25] studied the cycle embedding capability of THLNs with more faulty elements than the degree of the graph, i.e., under the large fault model. They The rest of this paper is organized as follows. Section 2 gives definitions and notions. Section 3 establishes the main result. Section 4 concludes the paper.
For basic graph-theoretic notations and terminology, the reader is referred to ref. [2]. For a graph G, let V (G) and E(G) denote its node set and edge set, respectively. For two nodes u
, and the degree of u in G is defined as that passes every node of the graph exactly once. A near-hamiltonian cycle (near-hamiltonian path, respectively) in a graph is a cycle (path, respectively) that passes every node but one of the graph exactly once.
For two nodes u and v in a graph G, let dist G (u, v) denote the distance between u and v, i.e., the minimum length of all paths between u and v. For a node x on a path P between u and v, if dist P (x, u) ≤ dist P (x, v), then we regard x as a u-closer node on P , and vice-versa.
According to [25], we give the definition of twisted hypercube-like network as follows.
Definition 2.1: For n ≥ 3, an n-dimensional (n-D, for short) twisted hypercube-like network (THLN, for short) is a graph G defined recursively as follows.
(1) For n = 3, G is isomorphic to the graph in Fig. 1.
(2) For n ≥ 4, G is constructed from two (n -1)-D THLN copies, G 1 and G 2 , in this way:
In what follows, we denote such a THLN as G = ⊕ φ (G 1 , G 2 ), and we use E c to denote the edge set u, φ(u) : u ∈ V (G 1 ) . The following important results on THLNs reported in [16], [17], [25] will be used in this paper. G -F contains a hamiltonian cycle if δ(G -F ) ≥ 2, and G -F contains a near-hamiltonian
For any two pairs of nodes [x 1 , x 2 ] and [y 1 , y 2 ] in G -F , there exist two paths P 1 and P 2 in G -F such that P 1 connects x 1 and y 1 , P 2 connects x 2 and y 2 , V (P 1 ) V (P 2 ) = ∅ and
This section deals with the fault-tolerant hamiltonian connectivity of THLNs under the large fault model. We can easily verify the following lemma.
there exists no path of length two or longer in G -F .
Excluding the above special cases, the main result of this paper is formulated as follows. Hence, the assertion is true for n = 7.
October 30, 2018 DRAFT Suppose the assertion holds for n
where G 1 and G 2 are k-D THLNs, E c = u, φ(u) : u ∈ V (G 1 ) . In the following discussion, we use a lowercase with subscript 1 to denote a node in V (G 1 ), and the same lowercase with subscript 2 to denote the node in V (G 2 ) such that these two nodes are connected by an edge in
, where |F | ≤ 2(k+1)-10 = 2k-8, and let
, and
for short), we may assume
The discussion will proceed by distinguishing the following five cases.
According to induction hypothesis, there exists a hamiltonian or near-hamiltonian path P 1 between s and t in G 1 -F 1 . We claim that we can find an edge (u 1 , v 1 ) on P 1 such that
). The
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