Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks THLNs

The twisted hypercube-like networks($THLNs$) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of $n$-dimensional($n$-$D$) $THLNs$. Let $G_n$ be an $n$-$D$ $THLN$ and $F$ be a subset of $V(G_n)\cup E(G_n)$ with $|F|\leq n-2$. We show that for arbitrary two different correct vertices $u$ and $v$, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^n-f_v-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_n-F$($n\geq5$).


Introduction
The n-dimensional hypercube [2], which possesses many outstanding properties such as recursive structure, relatively small degree, high symmetry, effective routing and broadcasting algorithms [3], is one of the most efficient, versatile interconnection network and, thus, becomes the preferred topological structure of parallel processing and parallel computing systems [4,5].Although hypercube networks have many excellent properties, it is well known that they also have inherent shortcomings, such as large diameter.Therefore, many scholars have proposed some hypercube variants, aiming at improving the defects of hypercubes, such as Efe's crossed cubes [10], Cull's and Larson's Mobius cubes [7], Hilbers's twisted cubes [9], Yang's locally twisted cubes [8].These hypercube variants retain the good properties of hypercubes, but also have many properties superior to hypercubes, such as the diameter of hypercube variants is almost half of the diameter of hypercubes.
The hypercube-like networks(short for HLNs) are a large class of network topologies [1,13,14].Among HLNs one may be identified as a subclass of networks, which in the paper is addressed as the twisted hypercube-like networks (short for T HLNs), proposed by Yang [31] in 2011.
Definition 1.1.[31] An n(n ≥ 3)-dimensional (short for n-D) twisted hypercube-like network (short for T HLN) is a graph defined recursively as follows.
(1) A 3-D T HLN is isomorphic to the graph depicted in Fig. 1(a).
(2) For n ≥ 4, an n-D T HLN G n is obtained from two vertex-disjoint (n − 1)-D T HLNs, denoted by G 0 n−1 and G 1 n−1 , in this way: ) is a bijective mapping.In the following, we will denote this graph G n as  Specifically, the fore-mentioned hypercube variant networks are all T HLNs.In 2005, Park et al. [13] demonstrated that all n-D T HLNs are Hamiltonian with at most n − 2 faulty elements and Hamiltonian connected with at most n − 3 faulty elements.Furthermore, Zhang et al. [15] improved the upper bound of fault tolerant Hamiltonian connectivity to n − 2 excepting only a pair of vertices and gave the definitions of weak vertex-pair and normal vertex-pair as follows.
Definition 1.2. [15]Let then w is called as a weak 2-degree vertex and (w 1 , w 2 ) is called as a w-weak vertex pair(short for weak vertex pair ).
If F = {a, b}, for instance, then w is a weak 2-degree vertex and (w 1 , w 2 ) is a weak vertex-pair in G 4 − F (See Figure 2).
Unquestionably, for the weak vertex-pair (w 1 , w 2 ), any correct path P w 1 w 2 of length l ≥ 3 can't include the weak 2-degree vertex w.It follows there is no correct hamiltonian path joining vertices w 1 and w 2 in G n −F [15].However, we proved that G n −F (n ≥ 5) contains at most one weak 2-degree vertex w and one w-weak vertex-pair for any [15].
Figure 2: Example of weak vertex-pair Definition 1.3. [15]If (w 1 , w 2 ) is not a weak vertex-pair for any vertex w ∈ V (G n −F ), then (w 1 , w 2 ) is addressed as a normal vertex pair.
In the paper, we studied the path-embedding in a T HLN with n−2 faulty elements and showed that if then for arbitrary two different correct vertices u and v, there is a fault-free path P uv of every length l with where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n − F (n ≥ 5).
To do this simply, we can denote For any vertex x ∈ L(or R), let x R (or x L ) be the sole vertex adjacent to vertex x in R(or L), and N L (x)(or N R (x)) be the set of vertices that are adjacent to vertex x in L(or R).Let E C be the set of edges that join L to R and E L (x)(or E R (x)) be the set of edges incident to vertex x in L(or R).
We use P uv to represent the path from vertex u to vertex v.
) and V (P uw ) ∩ V (P wv ) = {w}, we use P uw + P wv to denote the path ), P uv (u 1 , w 1 ) to represent the subpath of P uv which is from vertex u 1 to vertex w 1 , l uv to denote the length of P uv , d uv to denote the distance between vertex u to vertex v.We denote This paper is organized as below.Section 2 proved the main result.Section 3 concludes the paper.

Main Result
In the section, we will establish the main result of the paper.We depict theorem 2.1 as follows.
then for any two distinct fault-free vertices u and v, there exists a fault-free path P uv of every length l with where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n − F (n ≥ 5).

Proof.
We prove the theorem by the induction on n ≥ 5.The result holds for n = 5 by developing computer program using depth first searching technique combining with backtracking and branch and bound algorithm.Assume that the theorem holds for n − 1 with n ≥ 6, then we must show the theorem holds for n.In general, we assume Let u, v be any two distinct fault-free vertices in G n −F .By Theorem 1.1, there is a faultless path P uv of length l = 2 n − f v − 1 if vertices u and v form a normal vertex-pair in G n − F .Then we only need to find each length l with 2 n−1 − 1 ≤ l ≤ 2 n − f v − 2 between arbitrary different vertices u and v in G n − F .We divide the proof to two cases: (1) Since |F L | ≤ n − 3, by induction hypothesis, there is a faultless path , by a similar discussion, we can get a faultless path

Case 2. |F
We mark the faulty vertex x as faultless temporarily.Let By induction hypothesis, there is a faultless path P uv of each length l uv with 2 If the path P uv contains the faulty vertex x, let a, b ∈ N Puv (x); otherwise, we can arbitrarily select a vertex c from the path P uv .Let a, b ∈ N Puv (c).Since |F R | = 0, by induction hypothesis, there is a faultless path Case 2.1.
We mark the faulty vertex x as faultless temporarily.Let By induction hypothesis, there is a faultless path 6).By induction hypothesis, there is a faultless path In this paper, we apply our strategy to these four network topologies(MQ n , LT Q n , T Q n , CQ n ).In the future work, we will extend our strategy to other graphs of Hypercube-Like Networks.

Figure 1 :
Figure 1: Examples of 3D T HLN and 4D T HLN there is a faultless edge ab with ab ∈ E C , a, b / ∈ {u, v} and a, b / ∈ F .By induction hypothesis, there is a faultless path P ua of each length l ua with 2

Figure 3 :
Figure 3: Illustrations of proofs of Case 1.1 and Case 1.2 of Theorem 2.1.

Figure 4 :
Figure 4: Illustrations of proofs of Case 2.1.1 of Theorem 2.1.