Reliability Evaluation of Generalized Exchanged X-Cubes Based on the Condition of g-Good-Neighbor

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China Fujian Provincial Key Laboratory of Information Security of Network Systems, Fuzhou University, Fuzhou 350108, China College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117, China Fujian Provincial Key Laboratory of Network Security and Cryptology, Fujian Normal University, Fuzhou 350007, China Key Laboratory of Spatial Data Mining and Information Sharing, Ministry of Education, Fuzhou 350116, China

network can be used to measure the reliability of a multiprocessor system. In the following, we do not distinguish among multiprocessor systems, interconnection networks, and graphs.
An important evaluating parameter for the fault tolerance of a system (modeled by graph G), the connectivity, is denoted by κðGÞ, which is the minimum number of nodes that make the graph disconnected. So far, connectivities for many famous networks have been proven. Nevertheless, there is a shortcoming for using traditional connectivity as a parameter of fault tolerance, which is considered a highly unlikely phenomenon in reality that all nodes adjacent to a node have failed simultaneously. Therefore, Esfahanian and Hakimi [3] proposed a new measure to overcome this shortcoming, the restricted connectivity, which limits that all adjacent nodes of any node cannot fail at the same time. Later, a generalized restricted connectivity concept, the g-restricted connectivity (R g -connectivity), was proposed by Latifi et al. [4], which defines that each node of any remaining component after deleting all faulty nodes has degree at least g. In recent years, they have attracted much interest of theoretical computer scientists and mathematicians. Xu et al. [5] determined the R g -connectivity of hierarchical cubic networks and complete cubic networks. Ning [6] studied the R g -connectivity of exchanged crossed cubes. Yuan et al. [7] explored the R g -connectivity of κ-ary n-cube networks. Lin et al. [8] obtained the R g -connectivity of (n, k)-arrangement graphs.
Identifying all faulty processors in a multiprocessor system (in brief, system) is called system-level diagnosis. A system is t-diagnosable when all faulty processors can be detected, provided that the number of faulty processors in it does not exceed t. The maximum number of faulty processors that the system can precisely point out is as known as the diagnosability of the system. In system-level diagnosis, there are several well-known models.
The PMC model is the first model, proposed by Preparate et al. [9], which is a test-based model, assumes that the adjacent processors can perform tests on each other. For any adjacent processors in a system, the ordered pair hx, yi is called a test that x diagnoses its neighbor y, where x is a tester and y is a testee. In case x diagnoses y to be faulty (resp., faultfree), the outcome of the test hx, yi is 1 (resp., 0). Moreover, the outcome is reliable in the present of the tester x is faultfree. Another model, the MM model, was proposed by Maeng and Malek [10], which is a comparison-based model. In MM model, a comparator processor z sends the same test to its two neighbors x, y (i.e., comparison nodes) and then compares their responses. Let a labeled edge ðx, yÞ z be a comparison performed that two processors x and y are compared by a processor z, where x is adjacent to z and y is also adjacent to z. If the comparator processor z is fault-free, and the responses of x and y are identical, then both comparison processors x and y are fault-free; on the other hand, if the responses of x and y are different, then at least one of x, y is faulty. Furthermore, if both comparison processors x and y are faulty, the responses of x and y are distinct. In addition, the comparison ðx, yÞ z is unreliable in the present if the comparator node z is faulty. The MM * model (proposed by Sengupta and Dahbura) [11] is a special MM model, which is assumed that each processor must compare each pair of its adjacent processors.
Since there is no restrictive condition on the distribution pattern of faulty processors, the classical diagnosability of a system is quite small. In order to increase the diagnosability, Lai et al. [12] proposed a more realistic parameter of diagnosability, conditional diagnosability, which limited that all the neighbors of any processor cannot be faulty at the same time in a system. Recently, Peng et al. [13] proposed the notion of g-good-neighbor conditional diagnosability (g-GNCD), which is the maximum number of faulty processors that can be identified under the condition that every fault-free processor has no less than g fault-free neighbors. Peng et al. [13] (resp., Wang et al. [14]) established the g-GNCD of hypercubes under the PMC model (resp., MM * model). Li et al. [15] introduced the diagnosability and 1-goodneighbor conditional diagnosability of hypercubes with missing links and broken-down nodes under the PMC model. Yuan et al. [7] studied the g-good-neighbor conditional diagnosabilities of k-ary n-cube networks under the PMC model and the MM * model. Xu et al. [5] established the g-goodneighbor conditional diagnosabilities of complete cubic networks under the PMC model and the MM * model. Lin et al. [16] evaluated the g-good-neighbor conditional diagnosabilities of ðn, kÞ-arrangement graphs under the PMC model and the MM * model. Guo et al. [17] studied the g -good-neighbor conditional diagnosability of the crossed cubes under the PMC model and the MM * model. Li et al. [18] introduced this concept into a family of data center networks-DCell-and determined the g-good-neighbor conditional diagnosabilities of DCell under the PMC model and the MM * model.
The R g -connectivity (or g-GNCD) of different networks are usually determined independently. It is a very worthwhile topic to explore a unified method to get them in different networks. A family of exchanged networks (i.e., exchanged X-cubes) have some common properties, so that their R gconnectivity (or g-GNCD) can be studied by a uniform method. The family of exchanged X-cubes not only combine the advantages of hypercubes and some variant networks of hypercubes (X-cubes) but also reduce the interconnection complexity. Exchanged X-cubes classify its nodes into two different classes clusters with a unique connecting rule. In this paper, we propose the generalized exchanged X-cubes framework so that architecture can be constructed by different connecting rules. There are some of the better properties in generalized exchanged X-cubes, such as smaller diameter, fewer edges, lower cost factor, and low latency. Based on the fine properties, the network's hardware and communication costs are reduced, and a greater balance between performance and cost can be achieved. Due to the excellent properties of the generalized exchanged X-cubes, they can be used as the logical topologies in the peer-topeer environment [19].
In recent years, the research on the relationship between the R g -connectivity and the g-GNCD of regular networks under certain conditions has been widely developed [20][21][22][23][24], while this paper will study the R g -connectivity and the g-GNCD of a class of irregular networks (i.e., generalized 2 Wireless Communications and Mobile Computing exchanged X-cubes). We first establish the R g -connectivity of generalized exchanged X-cubes. Next, we evaluate the g-GNCD of generalized exchanged X-cubes. As applications, we obtain the R g -connectivity and g-GNCD of generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally generalized exchanged twisted cubes. The remainder of this paper is organized as follows. Section 2 provides the terms and notations used throughout the paper. Section 3 evaluates the R g -connectivity of generalized exchanged X-cubes. Section 4 establishes the g-GNCD of generalized exchanged X-cubes. Section 5 gives some applications based on the results in Section 3 and Section 4. In Section 6, we illustrate the advantages of R g -connectivity and g -GNCD compared to traditional connectivity and traditional diagnosability, respectively. Finally, we finish the whole paper by concluding in Section 7.

Preliminaries
2.1. Terminology and Notations. In this paper, a multiprocessor system is usually represented by a simple undirected graph (in brief, a graph). For terminology and notations not defined in this paper, we follow the reference [25]. We use G = ðVðGÞ, EðGÞÞ to represent a graph, where VðGÞ representing a nonempty and finite node set and EðGÞ = fðu, vÞ | ðu, vÞg is an unordered pair of VðGÞg representing an edge set. Two nodes u and v are adjacent, denoted by ðu, vÞ ∈ EðGÞ. The set of neighbors of node u in G is denoted by N G ðuÞ = fv ∈ VðGÞ | ðu, vÞ ∈ EðGÞg. If R ⊆ VðGÞ, let G½R denote the subgraph of G induced by the node subset R in G. And we denote G − R as G½VðGÞ \ R. We set N G ðRÞ = fv ∈ VðGÞ \ R | ðu, vÞ ∈ EðGÞ and u ∈ Rg = S u∈R N G ðuÞ \ R and N G ½R = N G ðRÞ ∪ R. Two binary strings u = u 1 u 0 and v = v 1 v 0 are pair related, denoted by u ∼ v, if and only if ðu, vÞ ∈ fð00, 00Þ, ð01, 11Þ, ð10, 10Þ, ð11, 01Þg. The case that u and v are not pair related is denoted by u ≁ v [26].
The degree of u in G is denoted by deg G ðuÞ = jN G ðuÞj. Let δðGÞ = min fdeg G ðuÞ | u ∈ VðGÞg, ΔðGÞ = max fdeg G ðuÞ | u ∈ VðGÞg. K n is defined as a complete graph with n nodes. A path P is a sequence of distinct nodes with any two consecutive nodes in P that are adjacent. We use G 1 ≅ G 2 to represent the graph G 1 is isomorphic to the graph G 2 . A component is defined as a maximally connected subgraph of a graph.
Definition 1 (see [27]). Let R ⊆ VðGÞ. R is called a node-cut if G − R is disconnected. If there exists a node-cut R with jRj = k, then R is called a k-node-cut. The connectivity κðGÞ of G is defined as the minimum k such that G has a k-node-cut.
Definition 2 (see [4]). Let g be a positive integer and R ⊆ VðGÞ. If G − R is disconnected and each remaining component has minimum degree at least g, then R is called an R g -cut.
Definition 3 (see [4]). The R g -connectivity of G, denoted by κ g ðGÞ, is the minimum cardinality over all R g -cuts of G.

The g-Good-Neighbor
Conditional Diagnosability. Under the PMC model and MM * model, we call the notation Ω as the syndrome of the system, which is defined as the set of all test (comparison) results in a system G, where test results are based on the PMC model and comparison results are based on the MM * model. Define a faulty set F, where ∀i ∈ F, i is a faulty processor. Let ΩðFÞ be the set of test (comparison) results which could be produced if F is the faulty node set. We use c F 1 and c F 2 to represent two distinct faulty sets of VðGÞ. In case Ωð c F 1 Þ ∩ Ωð c F 2 Þ = ∅, we call these two distinct faulty sets c F 1 and c F 2 distinguishable, and ð c F 1 , c F 2 Þ a distinguishable pair; otherwise, c F 1 and c F 2 are indistinguishable, In [28], under the PMC model, the sufficient and necessary condition for two different subsets c F 1 and c F 2 is a distinguishable pair proposed by Dahbura and Masson. Moreover, under the MM * model, the sufficient and necessary condition for two different subsets c F 1 and c F 2 is a distinguishable pair proposed by Sengupta and Dahbura [11].
Lemma 5 (see [11]). Let G = ðVðGÞ, EðGÞÞ be a multiprocessor system. For any two distinct sets c F 1 , c F 2 ⊆ VðGÞ, c F 1 and c F 2 are distinguishable under the MM * model if and only if there is a node w ∈ VðGÞ − c F 1 ∪ c F 2 such that one of the following conditions holds (see Figure 1(b)): The concept of g-GNCD of a system was proposed in the literature [13].
(1) LetF ⊆ VðGÞ andF be a fault-set. If any node of VðGÞ −F has at least g neighbors in G −F, thenF is called a g-good-neighbor conditional fault-set.
(2) A system G is g-good-neighbor conditional t-diagnosable if each distinct pair of g-good-neighbor conditional faulty (g-GNCF) sets c F 1 and c F 2 of VðGÞ with j c F 1 j ≤ t and j c F 2 j ≤ t are distinguishable.

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(3) The g-GNCD, denoted by t g ðGÞ, is defined as the maximum value of t such that G is g-good-neighbor conditionally t-diagnosable. Let t P g ðGÞ and t M g ðGÞ be the g-GNCD of G under the PMC model and MM * model, respectively.

Generalized Exchanged X-Cubes.
In this subsection, we give the definition of the family of generalized exchanged networks, denoted by generalized exchanged X-cubes, which have some common properties, so that the their R g -connectivity (or g-GNCD) can be studied by a uniform method. Since generalized exchanged X-cubes are derived by BC networks (bijective connection networks), we first review the definition of the BC network.
Definition 7 (see [29]). The one-dimensional BC network X 1 contains only two nodes which forms an edge. We use L 1 to represent the family of the one-dimensional BC network with L 1 = fX 1 g. A graph G belongs to the family of n-dimensional BC networks L n if and only if there exists V 0 , V 1 ⊂ VðGÞ such that the following two conditions hold: EðV 0 , V 1 Þ is a perfect matching M between V 0 and V 1 in G For any X n ∈ L n , by Definition 7, there exist V 0 , V 1 , M satisfying the conditions. We use X 0 n−1 , X 1 n−1 to denote the induced subgraph G½V 0 , G½V 1 , respectively. Clearly, they are both ðn − 1Þ-dimensional BC networks, and EðX 0 n−1 Þ, EðX 1 n−1 Þ, M is a decomposition of EðX n Þ. We define the decomposition as X n = GðX 0 n−1 , X 1 n−1 ; MÞ. BC networks are a class of networks containing a number of famous networks such as hypercubes [13], the Möbius cubes [30], crossed cubes [31], and locally twisted cubes [32] as members. An n-dimensional BC network X n is n-regular and consisting of 2 n nodes. Figure 2 shows two three-dimensional BC networks. Lemma 8 (see [33]). For 0 ≤ g ≤ n and Y ⊂ VðX n Þ, if δðX n ½YÞ ≥ g, then jYj ≥ 2 g . Lemma 9 (see [34]).
Lemma 10 (see [35]). For n ≥ 2, there are at most two common neighbors between any two nodes of X n . Next, we introduce the definition of generalized exchanged X-cubes.
Definition 11. The ðs, tÞ-dimensional generalized exchanged X-cubes is defined as a graph GEXðs, tÞ = ðVðGEXðs, tÞÞ, EðGEXðs, tÞÞÞ, for s ≥ 1 and t ≥ 1. GEXðs, tÞ consists of two disjoint subgraphs e L′ and e R′. And e L′ consists of 2 t subgraphs, Similarly, e R′ consists of 2 s subgraphs, denoted by f R j ′ for j = 1, 2, ⋯, 2 s . Moreover, GE Xðs, tÞ satisfies the following conditions (see Figure 3): there exists no edge between them. Similar for f R j ′ and f R k ′ with j ≠ k.
By Definition 11, we can deduce that jVðGEXðs, tÞÞj = 2 s+t+1 . Let each of f L i ′ and f R j ′ be a cluster of GEXðs, tÞ. Obviously, GEXðs, tÞ consists of 2 t + 2 s clusters. If we contract each cluster as a node, then GEXðs, tÞ is contracted into a complete bipartite graph K 2 t ,2 s . The edges that connect

The R g -Connectivity of GEXðs, tÞ
In this section, we establish the R g -connectivity of GEXðs, tÞ In what follows, we exploit some useful lemmas for our further investigation.

Lemma 12.
For any integers s ≥ 3 and 1 ≤ g ≤ s, let H be a subgraph of X s with δðHÞ ≥ g, and let T be a subgraph of X s such that T ≅ X g . Then jN X s ½Hj ≥ jN X s ½Tj = ðs − g + 1Þ2 g .
Proof. We conduct induction on s. If s = 3, by fixing g, the lemma holds obviously. Suppose that the lemma holds for s = τ − 1, let H be a subgraph of X τ−1 with δðHÞ ≥ g and T 1 be a subgraph of X τ−1 such that In the following, we will prove that the lemma holds for s = τ. Since X τ can be merged through a perfect matching by two X τ−1 , namely X 0 τ−1 and X 1 τ−1 , we discuss the two cases below.
Let T 2 and T 3 be two subgraphs of X 0 τ−1 and X 1 τ−1 with T 2 ≅ X g−1 and T 3 ≅ X g−1 , respectively. Thus, by the induction hypothesis, we have Then, for s = τ, the lemma holds.  Figure 2: Two three-dimensional BC networks. Without loss of generality, we suppose that H ⊆ X 0 τ−1 . By Lemma 8, we have jVðHÞj ≥ 2 g . Further, by the induction hypothesis, Hence, the lemma holds.
Proof. We assume U as a minimum R g -cut of GEXðs, tÞ. Let 1 ≤ j ≤ 2 s . Then we will show that κ g ðGEXðs, tÞÞ = jUj ≥ ðs − g + 1Þ2 g with 1 ≤ g ≤ s − 2 and s ≥ 3. We consider three cases as follows.
are connected for each i, j, We prove this case by contradiction. Suppose that jUj ≤ ðs − g + 1Þ2 g − 1. In the following, we will prove that U is not an R g -cut of GEXðs, tÞ.

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Without loss generality, assume that f is a ðg − 1Þ-good-neighbor cut of GEXðs, tÞ.

Case 3. For any integers
that are disconnected.

The g-Good-Neighbor Conditional Diagnosability of GEXðs, tÞ
In this section, we will determine the g-GNCD of GEXðs, tÞ under the PMC model and MM * model, respectively, where
Next, we prove that t P g ðGEXðs, tÞÞ ≥ ðs − g + 2Þ2 g − 1 with 1 ≤ g ≤ s − 2 and s ≥ 3. We suppose, to the contrary, that t P g ðGEXðs, tÞÞ ≤ ðs − g + 2Þ2 g − 2 for 1 ≤ g ≤ s − 2. And assume that there are two indistinguishable g-GNCF sets In what follows, we consider two cases.
Since c F 1 ≠ c F 2 , we may assume that c F 2 − c F 1 ≠ ∅. There exists no edge between VðGEXðs, tÞÞ − c F 1 ∪ c F 2 and c F 1 Δ c F 2 because c F 1 and c F 2 are indistinguishable. Moreover, since c F 1 is a g-good-neighbor conditional faulty set, it is easy to verify that δðGEXðs, tÞ½ c F 2 − c F 1 Þ ≥ g. By Lemma 8, j c F 2 − c F 1 j ≥ 2 g . On the other hand, since both c F 1 and c F 2 are g-GNCF sets, c F 1 ∩ c F 2 is also a g-good-neighbor conditional faulty set. Moreover, there is no edge between VðGEXðs, tÞÞ − c F 1 ∪ c F 2 and c F 1 Δ c F 2 ; thus, GEXðs, tÞ − c F 1 ∩ c F 2 is disconnected. Then c F 1 ∩ c F 2 is an R g -cut of GEXðs, tÞ. By Theorem 14, j c F 1 ∩ c F 2 j ≥ ðs − g + 1Þ2 g with s ≥ 3 and 1 ≤ g ≤ s − 2. Hence, which results in a contradiction since j c F 2 j ≤ ðs − g + 2Þ2 g − 1.
Hence, the theorem holds.
Proof. The proof of t M g ðGEXðs, tÞÞ ≤ ðs − g + 2Þ2 g − 1 with 1 ≤ g ≤ s − 2 and s ≥ 4 is similar to Theorem 15, so it is omitted.
Thus, VðGEXðs, tÞÞ ≠ c F 1 ∪ c F 2 . In addition, an important claim is given as follows.
By contradiction, suppose that GEXðs, tÞ − c F 1 ∪ c F 2 has at least one isolated node. Then, we prove that the two cases both contradict the supposition. 8 Wireless Communications and Mobile Computing Case 1. g = 1.
Since c F 1 ≠ c F 2 , without loss of generality, we suppose that Xðs, tÞ − c F 1 ∪ c F 2 has no isolated node. Now, we consider c F 1 ⊈ c F 2 . The given W is the set of all isolated nodes and Since c F 1 and c F 2 are indistinguishable, there exists at most one node u ∈ c F 2 − c F 1 with u is adjacent to w by Lemma 5. Thereby, there exists only one node u ∈ c F 2 − c F 1 with u adjacent to w. It is easy to see that there is only It follows that jWj ≤ 2sðt + 1Þ/t − 1 ≤ 4s. Thus, Let f ðsÞ = 2 2s − 7s − 3. We can deduce that ∂f ðsÞ/∂s > 0. Then f ðsÞ is an increasing function. Therefore, f ðsÞ ≥ f ð4Þ > 0, a contradiction. Thus, VðBÞ ≠ ∅.
Since the fault-pair ð c F 1 , c F 2 Þ does not satisfy Lemma 5 and any node in VðBÞ is not isolated, there exists no edge between VðBÞ and c Xðs, tÞÞ and ðu, wÞ ∈ EðGEXðs, tÞÞ for any isolated node w ∈ W. By Lemma 10, there are at most two common neighbors between any two nodes in VðX s Þ. In addition, by Definition 11, any two cross edges have no common end node. Then we deduce that any two nodes in VðGEXðs, tÞÞ have at most two common neighbors. Thus, jWj ≤ 2. Since there is no common node between any two cross edges and X s is triangle-free, GEXðs, tÞ is triangle-free. Thereby, Therefor, for s ≥ 4, we have which results in a contradiction since j c F 2 j ≤ 2s + 1.
Without loss of generality, we suppose that c F 2 − c F 1 ≠ ∅. Since c F 1 is a g-GNCF set of GEXðs, tÞ, jN GEXðs,tÞ− b Since w ∈ VðGEXðs, tÞ − c F 1 ∪ c F 2 Þ is arbitrary, every node of GEXðs, tÞ − c F 1 ∪ c F 2 is not an isolated one.
To sum up, Claim 25 holds.
Since there exists no isolated node in GEXðs, tÞ − c F 1 ∪ c F 2 by Claim 25 we have, for any w ∈ GEXðs, tÞ − c F 1 ∪ c F 2 , there exists some node v ∈ GEXðs, tÞ − c F 1 ∪ c F 2 such that ðw, vÞ ∈ EðGEXðs, tÞÞ. If ðu, wÞ ∈ EðGEXðs, tÞÞ for any u ∈ c F 1 Δ c F 2 , ðu, vÞ w satisfies condition in Lemma 5. Therefore, the g-GNCF sets c F 1 and c F 2 are distinguishable, which results in a contradiction. By the arbitrariness of w ∈ GEXðs, tÞ − c F 1 ∪ c F 2 , there exists no edge between VðGEXðs, tÞÞ − c F 1 ∪ c F 2 and c F 1 △ c F 2 .
Since c F 1 is a g-GNCF set and c Since c F 1 and c F 2 are both g-GNCF sets and there exists no edge between Thus, t M g ðGEXðs, tÞÞ ≥ ðs − g + 2Þ2 g − 1 for any integers 1 ≤ g ≤ s − 2 and s ≥ 4.
Hence, the proof of theorem is completed.

Applications to a Family of Famous Networks
In Section 2, the definition of the generalized exchanged Xcube GEXðs, tÞ has been given. Furthermore, we determine the R g -connectivity and g-GNCD of GEXðs, tÞ in Section 3 and Section 4, respectively. Applying the theorems of 9 Wireless Communications and Mobile Computing Section 3 and Section 4, we can directly establish the R gconnectivity and g-GNCD of some generalized exchanged X-cubes, including generalized exchanged hypercubes, dualcube-like networks, generalized exchanged crossed cubes, and locally generalized exchanged twisted cubes. In this section, we will give the applications to these networks.

The Generalized Exchanged
Hypercube. In 2005, Loh et al. [36] proposed the exchanged hypercube, which obtained by removing edges from a hypercube H s+t+1 . We denote I r = f1, 2,⋯,rg, where r is a given position integer. For each n ∈ I r , the sequence x r x r−1 ⋯ x 1 is a binary string of length r if x n ∈ f0, 1g. The definition of exchanged hypercubes is presented as follows.
Definition 16 (see [36]). Let s, t ≥ 1, the exchanged hypercube EHðs, tÞ consists of the node set VðEHðs, tÞÞ and the edge set EðEHðs, tÞÞ, two nodes u = u s+t ⋯ u t+1 u t ⋯ u 1 u 0 and v = v s+t ⋯ v t+1 v t ⋯ v 1 v 0 are linked by an edge, called r-dimensional edge, if and only if the following conditions are satisfied: (i) u and v differ exactly in one bit on the r-th bit or on the last bit The generalized exchange hypercube was proposed by Cheng et al. [37]. Let s, t ≥ 1, the generalized exchanged hypercube GEHðs, t, f Þ consists of two classes of hypercubes: one class contains 2 t H s 's, referred to as the Class-0 clusters; and the other contains 2 s H t 's, referred to as the Class-1 clusters. Class-0 and Class-1 clusters will be referred to as clusters of opposite class of each other, same class otherwise. The function f is a bijection between nodes of Class-0 clusters and those of Class-1 clusters; for two nodes u, v in the same cluster, f ðuÞ and f ðvÞ are in two different clusters, and the edge ðu, f ðuÞÞ is a cross edge. The bijection f ensures the existence of a perfect matching between nodes of Class-0 clusters and those in the Class-1 clusters but ignores the specifics of the perfect matching. Hence, we present the following proposition.  Figure 5 shows the GEHð1, 1Þ and GEHð1, 2Þ): there exists no edge between them. Similar for f R j ′ and f R k ′ with j ≠ k.
The dual-cube is a special case of the exchanged hypercube when s = t, proposed by Li and Peng [38]. That is, EHðn, nÞ ≅ D n . The dual-cube-like network DC n [39], which is a generalization of dual-cubes, is isomorphic to EHðn − 1, n − 1Þ, a special case of GEHðn − 1, n − 1Þ (see DC 3 in Figure 6).
By Proposition 17, the generalized exchanged hypercube GEHðs, tÞ is the member of generalized exchanged X-cubes, where the X-cube is a hypercube. Then, the following theorems hold obviously.
5.2. The Generalized Exchanged Crossed Cube. Li et al. [26] give the definition of ECQðs, tÞ, which is obtained by removing edges from a crossed cube CQ s+t+1 . In what follows, we review the definition of exchanged crossed cubes.
where ⊕ is the exclusive-OR operator.
Let s, t ≥ 1, the generalized crossed cube GECQðs, t, f Þ comprises two classes of crossed cubes, referred to as the Class-0 clusters and the Class-1 clusters, respectively. The Class-0 clusters contain 2 t CQ s 's and the Class-1 clusters contain 2 s CQ t 's. They will be referred to as clusters of opposite class of each other, same class otherwise. The function f is a bijection between nodes of Class-0 clusters and those of Class-1 clusters such that, for u, v, two nodes of the same cluster, f ðuÞ and f ðvÞ, are in two different clusters, and the edge ðu, f ðuÞÞ is a cross edge. The bijection f ensures the existence of a perfect matching between two nodes in different clusters, but there is no requirement for the specifics of the perfect matching. Therefore, we have the following proposition.

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with 0 ≤ j ≤ s − 1 and 0 ≤ i ≤ t − 1g.
By Definition 21, e L ′ can be partitioned into 2 t sub- Similarly, e R′ can be partitioned into 2 s subgraphs, denoted by f R j ′ such that w 1 , w 2 ∈ f R j ′, w 1 ½t + 1 : s + t = w 2 ½t + 1 : s + t, for j = 1, 2, ⋯, 2 s . And GECQðs, tÞ satisfies the following conditions (see GECQð1, 3Þ in there exists no edge between them. Similar for f R j ′ and f R k ′ with j ≠ k.
5.3. The Locally Generalized Exchanged Twisted Cube. The locally exchanged twisted cube proposed by Chang et al. [29], obtained by removing edges from a locally twisted cube LTQ s+t+1 . The definition of locally exchanged twisted cube is introduced as follows.
Let s, t ≥ 1; there are two classes of locally twisted cubes in the locally generalized exchanged twisted cube LGETQ ðs, t, f Þ: one class, referred to as the Class-0 clusters, contains 2 t LTQ s 's; and the other, referred to as the Class-1 clusters, contains 2 s LTQ t 's. They will be referred to as clusters of opposite class of each other, same class otherwise. There exists a bijection function f between nodes of Class-0 clusters and those of Class-1 clusters. For two nodes u, v in the same cluster, f ðuÞ and f ðvÞ belong to two different ones, and the edge ðu, f ðuÞÞ is a cross edge. The bijection f ensures the existence of a perfect matching between nodes of Class-0 clusters and those in the Class-1 clusters, but the specifics of the perfect matching can be ignored. Further, we obtain the proposition as follows.

Proposition 26.
LGETQðs, tÞ can be decomposed into two disjoint subgraphs e L ′ and e R ′ . e L ′ can be partitioned into 2 t subgraphs, denoted by f L i ′ for i = 1, 2, ⋯, 2 t . Similarly, e R ′ can be partitioned into 2 s subgraphs, denoted by f R j ′ for j = 1, 2, ⋯, 2 s . And LGETQðs, tÞ satisfies the following conditions (see L GETQð1, 3Þ in there exits no edge connects them. Similar for f R j ′ and f R k ′ with j ≠ k.