The extra connectivity and extra diagnosability of regular interconnection networks

Journal Pre-proof The extra connectivity and extra diagnosability of regular interconnection networks

Mengjie Lv, Jianxi Fan, Jingya Zhou, Baolei Cheng, Xiaohua Jia

PII:

S0304-3975(19)30763-7

DOI:

https://doi.org/10.1016/j.tcs.2019.12.001

Reference:

TCS 12292

To appear in:

Theoretical Computer Science

Received date:

22 October 2019

Accepted date:

1 December 2019

Please cite this article as: M. Lv et al., The extra connectivity and extra diagnosability of regular interconnection networks, Theoret. Comput. Sci. (2020), doi: https://doi.org/10.1016/j.tcs.2019.12.001.

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The extra connectivity and extra diagnosability of regular interconnection networks Mengjie Lv1 , Jianxi Fan1∗ , Jingya Zhou1 , Baolei Cheng1 , Xiaohua Jia2 , 1 School of Computer Science and Technology, Soochow University, Suzhou 215006, China 2 Department of Computer Science, City University of Hong Kong, Hong Kong ∗ Corresponding author: [email protected] Abstract: Reliability assessment of interconnection networks is critical to the design of multiprocessor systems. Extra connectivity and extra diagnosability are two important metric parameters for the reliability evaluation of interconnection networks. In this paper, we first establish that the i-extra connectivities of regular networks with some conditions Gn are (i + 1)n − 2i for i = 1, 2, where n ≥ 4, and then determine that the i-extra diagnosabilities of the networks under the PMC model and the MM* model are (i + 1)n − i for i = 1, 2, where n ≥ 4. Furthermore, two polynomial time diagnostic algorithms of regular interconnection networks under the PMC and MM* model are described. Finally, the corresponding extra connectivities and extra diagnosabilities of some networks, including the star networks, the pancake networks and the burnt pancake networks, can be derived by our results. Keywords: Interconnection networks; extra connectivity; extra diagnosability; PMC model; MM* model. 1. Introduction With the rapid development of high performance computing technology, there are more and more processors in parallel computer interconnection networks, which makes it inevitable for processors in the interconnection networks to fail. Reliability has always been a key issue in interconnection networks. The reliability of interconnection networks can be characterized by two parameters – connectivity and diagnosability. In the following, we do not distinguish between interconnection networks and networks. An interconnection network topology is usually represented by a graph G(V, E), where vertices represent processors and edges represent links between processors. The connectivity of a graph G, denoted by κ(G), which is the size of a minimal set of vertex whose removal makes the network disconnected or becomes a trivial graph, is the key graph-theoretic concept to measure the reliability of the interconnection networks. The higher the connectivity, the more reliable the network. By Menger’s theorem, if the connectivity of a graph G is n, then there exist n disjoint paths between any two distinct vertices in G. As a result, when sending data from a vertex x to a vertex y in G, x will have n available paths that can be used to transmit data. This means that there always exists one fault-free path between x and y even if up to n − 1 faulty vertices occur in G. Nevertheless, the connectivity is no 1

more than the minimum node-degree of the network, which means that the network will have a lower reliability under the concept of connectivity. In order to increase the reliability of network, researchers have proposed various connectivities by adding restricted conditions to faulty vertex set [13, 18, 22, 33, 37]. In particular, F`abrega and Fiol [7] proposed the g-extra connectivity, which is the minimum cardinality of vertices whose deletion will disconnect the network and each residual component has more than g vertices. The extra connectivity has many applications in some known networks, such as hypercube-like networks (also called bijective graphs) [2, 8, 41], k-ary n-cubes [9, 14], alternating group networks [32], bubble-sort star networks [11], data center networks [20]. We call the process of identifying faulty processors through analyzing the results of mutual tests among processors to be system-level diagnosis. Two famous models have been put forward with various diagnosis strategies. In 1967, Preparata et al. [28] proposed a testbased diagnostic method, namely PMC model, in which each test is conducted between two adjacent processors, and it was supposed that a test result is reliable (resp., unreliable) if the tester is fault-free (resp., faulty). Subsequently, Malek and Maeng [27] used comparisonbased diagnosis method to describe the another fault diagnostic model – MM model, under which the diagnosis is performed by sending the same inputs to a pair of adjacent processors, and then comparing their responses. Sengupta and Dahbura [29] further generalized the MM model and introduced the MM* model, where each processor will test another two processors if it is adjacent to them. Numerous studies have been focused on PMC model and MM* model [4, 21, 35, 39, 40] A system is t-diagnosable if all faulty processors can be identified without a replacement, supposed that the number of occurring faults does not exceed t. The diagnosability of a system, denoted by t(G), is the maximal number of faulty vertices that can be diagnosed by the system. Owing to the unpractical assumption that t(G) mandatorily demands that all the adjacent processors of a processor failing at the same time, the classical diagnosability of a system is quite small. In view of above, Lai et al. [17] developed a new measure of diagnosability, named as conditional diagnosability, which assumes that all adjacent vertices of any vertex cannot be faulty simultaneously. To better reflect a network’s real diagnosability and system-level diagnostic ability, Zhang and Yang [38] generalized the concept of conditional diagnosability and proposed the g-extra diagnosability, denoted by t˜g (G), which is the maximum number of faulty vertices that the system can guarantee to identify under the condition that every fault-free component has more than g vertices. The g-extra diagnosability of the system has received much attention [20, 23, 25, 26, 31, 32]. Most of the above studies on connectivity and diagnosability are based on specific networks. However, the characterization of reliability for general regular network is more meaningful than that of special network. Recently, the {1, 2}-good-neighbor diagnosability of some regular networks were obtained by Wei et al. [34]. Gu et al. [10] gave an equal relation between the 1-extra connectivity and pessimistic diagnosability for some regular networks. Hao et al. [12] investigated the relationship between 2-extra connectivity and conditional diagnosability of regular networks under the MM* model. So far, there is little research on i-extra diagnosability of networks under two classical diagnostic models, especially MM* model. Thus, our goal is to establish the {1, 2}-extra connectivities of regular networks and determine the {1, 2}-extra diagnosabilities of regular networks under PMC model and MM* model. 2

In this paper, we aim to explore a general technical method to determine the extra connectivity and the extra diagnosability of regular networks with some conditions. The major contributions of this paper are listed as follows: (1) We show that the i-extra connectivities of Gn are (i + 1)n − 2i for i = 1, 2, where n ≥ 4. Once the {1, 2}-extra connectivities of a network are obtained, the corresponding conditional diagnosability and pessimistic diagnosability of the network can be determined immediately. (2) We determine that the i-extra diagnosabilities of Gn under the PMC model and the MM* model are (i + 1)n − i for i = 1, 2, where n ≥ 4. In addition, we give two polynomial time diagnostic algorithms of the regular interconnection networks under the PMC model and the MM* model. (3) As applications, we address the {1, 2}-extra connectivities and {1, 2}-extra diagnosabilities of some networks, including the star networks, the pancake networks and the burnt pancake networks. The rest of this paper is organized as follows. Section 2 introduces terminologies and preliminaries used throughout this paper. Section 3 explores the {1, 2}-extra connectivities of regular networks. Section 4 establishes the {1, 2}-extra diagnosabilities of regular networks under PMC model and MM* model, respectively, and gives two polynomial time diagnostic algorithms. As empirical analysis, the corresponding extra connectivities and extra conditional diagnosabilities of some networks, including the star networks, the pancake networks and the burnt pancake networks, are obtained in Section 5. Section 6 concludes the paper. 2. Preliminaries 2.1. Terminologies and Notations We use a graph G = G(V, E) to represent an interconnection network, where a vertex u ∈ V represents a processor and an edge (u, v) ∈ E represents a link between vertices u and v. For any vertex v of the graph G = (V, E), the neighborhood NG (v) of vertex v in G is defined as the set of all vertices which are adjacent to v, i.e., NG (v) = {u ∈ V | (u, v) ∈ E}. We also denote, by |NG (u)|, the degree d(u) of u. Let S be a subset of V , the subgraph of G induced by S, denoted by G[S], is the graph with the vertex-set S and the edge-set {(u,  v) | (u, v) ∈ E, u, v ∈ S}. We define NG (S) = {v ∈ V \ S | ∃u ∈ S, (u, v) ∈ E} = ( u∈S NG (u)) \ S and NG [S] = NG (S) ∪ S. When G is clear from the context, we use N (v) to replace NG (v), N (S) to replace NG (S), and N [S] to replace NG [S]. For convenience, we set n := {1, 2, . . . , n} in the context. For any subset F ⊂ V (G), the notation G − F denotes a graph obtained by removing all vertices in F from G and deleting those edges with at least one end-vertex in F , simultaneously. If there exists a nonempty subset F ⊂ V (G) such that G − F is disconnected, then F is called a vertex-cut, or separating set of G. The maximal connected subgraphs of G − F are called components. Given two sets of nodes F1 , F2 ⊂ V (G), the symmetric difference of two subsets F1 and F2 is defined as F1 F2 = (F1 − F2 ) ∪ (F2 − F1 ). A path in a graph is a sequence of distinct vertices such that there is an edge joining consecutive vertices, with the length being the number of edges. If a path has length l, then the path is called as an l-path. The girth of G, denoted by g(G), is the length of a shortest cycle of G. A matching of size k in a graph G is a set of k pairwise disjoint edges. The 3

vertices belonging to the edges of a matching are saturated by the matching. The others are unsaturated. A perfect matching is a matching that saturates every vertex of G. 2.2. Faulty vertex diagnosis and its determination Definition 1 [38] A vertex subset F of graph G is a g-extra vertex subset if and only if every component of G − F has more than g vertices. Definition 2 [38] A system G = (V, E) is g-extra t-diagnosable if and only if for each pair of distinct faulty g-extra vertex subsets F1 , F2 ⊆ V (G) such that |Fi | ≤ t (i = 1, 2), F1 and F2 are distinguishable. The g-extra diagnosability of G, denoted as t˜g (G), is the maximum value of t such that G is g-extra t-diagnosable. The PMC model is a basic diagnostic model in fault diagnosis theory. Under this model, the test-set is expressed by a digraph D(V, L), in which V represents the vertex-set of G and L represents the ordered edge-set. The arc (u, v) ∈ L represents that u tests v. Let σ(u, v) represent the outcome that u tests v. For any (u, v) ∈ L, ⎧ ⎨ 0, u ∈ F, v ∈ F ; 1, u ∈ F, v ∈ F ; σ(u, v) = ⎩ 0/1, u ∈ F, where F is the fault-set. Lemma 3 [28] For any two distinct subsets F1 and F2 in a system G = (V, E), the sets F1 and F2 are distinguishable under the PMC model if and only if there exist a vertex u ∈ V − (F1 ∪ F2 ) and a vertex v ∈ F1 F2 such that (u, v) ∈ E. The MM* model is also a basic diagnostic model in fault diagnosis theory. Under this model, the comparison scheme of a system G = (V, E) is often modeled as a multigraph M (V, L), where V and L are the vertex-set of G and the labeled edge-set, respectively. If both of vertices u and v are adjacent to w, then (u, v; w) ∈ L, which implies that u and v are being compared by w. Let σ(u, v; w) represents the test outcome. For each (u, v; w) ∈ L, ⎧ ⎨ 0, u ∈ F, v ∈ F, w ∈ F ; 1, {u, v} ∩ F = ∅, w ∈ F ; σ((u, v; w)) = ⎩ 0/1, w ∈ F, where F is the fault-set. Lemma 4 [29] For any two distinct subsets F1 and F2 in a system G = (V, E), the sets F1 and F2 are distinguishable under the MM* model if and only if one of the following conditions is satisfied. (1) There are two vertices u, w ∈ V − (F1 ∪ F2 ) and there is a vertex v ∈ F1 ΔF2 such that (u, w) ∈ E and (v, w) ∈ E; (2) There are two vertices u, v ∈ F1 − F2 and there is a vertex w ∈ V − (F1 ∪ F2 ) such that (u, w) ∈ E and (v, w) ∈ E; (3) There are two vertices u, v ∈ F2 − F1 and there is a vertex w ∈ V − (F1 ∪ F2 ) such that (u, w) ∈ E and (v, w) ∈ E. 4

3. The i-extra connectivity of regular networks In this section, we will explore the extra connectivity of regular networks Gn (V (Gn ), E(Gn )). Throughout this paper, let Gn = (V (Gn ), E(Gn )) be an n-connected network, in which G1 ∼ = K2 . For n ≥ 2, Gn consists of m (≥ n) disjoint (n − 1)-connected copies of m  Gn−1 , denoted by G1n−1 , G2n−1 , . . . , and Gm , where V (G ) = V (Gin−1 ) and E(Gn ) = n n−1 m  i=1

E(Gin−1 )



 1≤i=j≤m

i=1

E(i, j) with E(i, j) denoting the set of edges between Gin−1 and Gjn−1

for any 1 ≤ i = j ≤ m, such that the following conditions hold. (1) g(Gn ) ≥ 6 for n ≥ 2. (2) E(i, j) is a perfect matching of Gn . 1≤i=j≤m

Obviously, Gn is n-regular. For any vertex u ∈ V (Gin−1 ) with 1 ≤ i ≤ m, u has exact one neighbor u ∈ V (Gn − Gin−1 ), which is called as the external neighbor of u. Lemma 5 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 2n for n ≥ 2. Then, for any subset S ⊆ V (Gn ) with |S| = 2, |N (S)| ≥ 2n − 2. Proof Without loss of generality, we suppose that S = {u, v}. According to the distribution of u and v, we distinguish the following two cases. Case 1. |E(Gn [S])| = 0. That is, u and v are not adjacent. Noting that |N (u)∩N (v)| ≤ 1, we have |N (S)| ≥ 2n−1. Case 2. |E(Gn [S])| = 1. That is, u and v are adjacent. Since g(Gn ) ≥ 6, |N (u) ∩ N (v)| = 0. It follows that |N (S)| = 2(n − 1) = 2n − 2.  Lemma 6 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 3n − 1 for n ≥ 2. Then, for any subset S ⊆ V (Gn ) with |S| = 3, |N (S)| ≥ 3n − 4. Proof Without loss of generality, we suppose that S = {u, v, w}. According to the distribution of u, v and w, we distinguish the following three cases. Case 1. |E(Gn [S])| = 0. That is, u, v and w are not adjacent each other. Noting that g(Gn ) ≥ 6, we have |N (S)| ≥ 3n − 32 = 3n − 3. Case 2. |E(Gn [S])| = 1. That is, two vertices are adjacent to each other and the other vertex is isolated. Without loss of generality, we suppose that u and v are adjacent. Considering that g(Gn ) ≥ 6, there exists at most one common neighbor between either the pair of u and w, or the pair of v and w (i.e., |N (v)∩N (w)| = 0 or |N (u)∩N (w)| = 0). Thus, we have |N (S)| ≥ 3n−2−1 = 3n−3. Case 3. |E(Gn [S])| = 2. That is, Gn [S] consists of a 2-path, say P = u, v, w. By the facts that any two vertices have at most one common neighbor and g(Gn ) ≥ 6, we have |N (S)| = n − 1 + n − 2 + n − 1 = 3n − 4. 

5

For the convenience of the following discussion, we give some notations, which are used throughout this paper. Let F be a vertex-subset of Gn with Fi = F ∩ Gin−1 for any i ∈ m. We also denote I = {i ∈ m | |Fi | ≥ n − 1}, J = m − I, FI = ∪i∈I Fi , FJ = ∪j∈J Fj , and

GIn−1 = ∪i∈I Gin−1 , GJn−1 = ∪j∈J Gjn−1 .

Lemma 7 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 2n and |E(i, j)| > 2n − 4 for n ≥ 3. Then, for any subset F ⊆ V (Gn ) with |F | ≤ 2n − 3, Gn − F satisfies one of the following conditions. (1) Gn − F is connected. (2) Gn − F has exactly two components, one of which is an isolated vertex. Proof Considering that |F | ≤ 2n − 3, we have |I| ≤ 1 (otherwise, 2(n − 1) ≤ 2|Fi | ≤ |F | ≤ 2n − 3, a contradiction). Then |J| = m − |I| ≥ m − 1. For any j ∈ J, |Fj | ≤ n − 2 implies that Gjn−1 \Fj is still connected since Gjn−1 is isomorphic to Gn−1 whose connectivity is n − 1. 1 2 Since |E(j1 , j2 )| > 2n − 4, Gjn−1 is connected to Gjn−1 for any j1 , j2 ∈ J, which implies that J Gn−1 − FJ is connected. Let H be the union of small components of Gn −F that contains no vertex in GJn−1 −FJ . In the following, we will show that |V (H)| ≤ 1. Suppose, by contradiction, that |V (H)| ≥ 2. Let H  ⊆ V (H) with |H  | = 2. Then, we have NGin−1 (H  ) − V (H) ⊆ Fi and NGn −Gin−1 (V (H)) ⊆ F − Fi (see Fig. 1). By Lemma 5, we have

FJ

Fi

H

GnJ FJ

H

Gni

GnJ

Fig. 1: An illustration of the proof of Lemma 7. |Fi | ≥ NGin−1 (H  ) − (|V (H)| − 2) ≥ 2(n − 1) − 2 − (|V (H)| − 2) = 2n − 2 − |V (H)|.

6

However, since each vertex has exact one external neighbor, we have |Fi | ≤ |F | − |NGn −Gin−1 (V (H))| ≤ |F | − |V (H)| ≤ 2n − 3 − |V (H)|, a contradiction. Thus, the lemma holds.



Lemma 8 Let Gn = (V (Gn ), E(Gn )) be an n-connected network |V (Gn )| > 3n − 1 and |E(i, j)| > 2n − 4 for n ≥ 3. Then, for any subset F ⊆ V (Gn ) with |F | ≤ 3n − 5, Gn − F satisfies one of the following conditions. (1) Gn − F is connected. (2) Gn − F has two components, one of which is an isolated vertex or an isolated edge. (3) Gn − F has three components, two of which are isolated vertices. Proof Noting that |F | ≤ 3n − 5, we have |I| ≤ 2 (otherwise, 3(n − 1) ≤ 3|Fi | ≤ |F | ≤ 3n − 5, a contradiction). Then |J| = m − |I| ≥ m − 2. Similarly, GJn−1 − FJ is connected. Let H be the union of small components of Gn − F that contains no vertex in GJn−1 − FJ . In the following, we will show that |V (H)| ≤ 2. We distinguish the following two cases. Case 1. |I| = 1. Suppose, by contradiction, that |V (H)| ≥ 3. Let H  ⊆ V (H) with |H  | = 3. Then we have NGin−1 (H  ) − V (H) ⊆ Fi and NGn −Gin−1 (V (H)) ⊆ F − Fi . By Lemma 6, we have |Fi | ≥ NGin−1 (H  ) − (|V (H)| − 3) ≥ 3(n − 1) − 4 − (|V (H)| − 3) = 3n − 4 − |V (H)|. However, since each vertex has at most one external neighbor, we have |Fi | ≤ |F | − |NGn −Gin−1 (V (H))| ≤ |F | − |V (H)| ≤ 3n − 5 − |V (H)|, a contradiction. Case 1. |I| = 2. Without loss of generality, we suppose that I = {i, j} and |Fj | ≤ |Fi | ≤ |F | − |Fj | ≤ 3n−5−(n−1) = 2n−4. Thus, we have |FJ | = |F |−|Fi |−|Fj | ≤ 3n−5−(n−1)−(n−1) = n−3. It follows that GJn−1 − FJ is connected. By Lemma 7, Gin−1 − Fi is either connected or has exactly two components, one of which is an isolated vertex. If Gin−1 − Fi is connected, then Gin−1 − Fi is connected to GJn−1 − FJ and |V (H) ∩ Gin−1 | = 0 because |E(x, y)| > 2n − 4 for any x = y ∈ {1, 2, . . . , m}. If there exists an isolated vertex, say u, in Gin−1 − Fi . Then Gin−1 − Fi − {u} is connected to GJn−1 − FJ and |V (H) ∩ Gin−1 | ≤ 1 because |E(x, y)| > 2n − 4 for any x = y ∈ {1, 2, . . . , m}. Similarly, we can obtain |V (H) ∩ Gjn−1 | ≤ 1. Thus, we have |V (H)| ≤ 2. Hence, the lemma holds.  7

Theorem 9 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 2n and (1) |E(i, j)| > 2n − 4 for n ≥ 3. Then, the 1-extra connectivity of Gn is κo (Gn ) = 2n − 2. (1)

(1)

Proof First, we show that κo (Gn ) ≥ 2n − 2. Suppose, to the contrary, that κo ≤ 2n − 3. (1) Let F be a minimum 1-extra cut of Gn . Then |F | = κo ≤ 2n − 3. According to Lemma 7, Gn − F has a small component M with |M | ≤ 1, a contradiction. (1) Next, we show that κo (Gn ) ≤ 2n − 2. Let (u, v) ∈ E(Gn ) and F = NGn ({u, v}). Then, |F | = 2n − 2. Noting that |NGn [{u, v}]| = 2n − 2 + 2 = 2n < |V (Gn )|, we have V (Gn ) − {u, v} − F = ∅, which implies that F is a vertex cut. Since g(Gn ) ≥ 6, |NGn (x) ∩ NGn ({u, v})| ≤ 1 for any vertex x ∈ V (Gn ) − {u, v} − F . Thus, we have dGn −F (x) ≥ n − 1 ≥ 2, (1)

which implies that F is a 1-extra vertex cut. Hence, κo (Gn ) ≤ 2n − 2.



Theorem 10 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 3n − 1 (2) and |E(i, j)| > 2n − 4 for n ≥ 4. Then, the 2-extra connectivity of Gn is κo (Gn ) = 3n − 4. (2)

(2)

Proof First, we show that κo (Gn ) ≥ 3n − 4. Suppose, to the contrary, that κo ≤ 3n − 5. (2) Let F be a minimum 2-extra cut of Gn . Then |F | = κo ≤ 3n − 5. According to Lemma 8, Gn − F has a small component M with |M | ≤ 2, a contradiction. (2) Next, we show that κo (Gn ) ≤ 3n − 4. Let u, v, w ∈ V (Gn ) and (u, v), (v, w) ∈ E(Gn ) and let F = NGn ({u, v, w}). Then, |F | = 3n − 4. Noting that |NGn [{u, v, w}]| = 3n − 4 + 3 = 3n − 1 < |V (Gn )|, we have V (Gn ) − {u, v, w} − F = ∅, which implies that F is a vertex cut. Since g(Gn ) ≥ 6, |NGn (x) ∩ NGn ({u, v, w})| ≤ 2 for any vertex x ∈ V (Gn ) − {u, v} − F . Thus, we have (2) dGn −F (x) ≥ n − 2 ≥ 2. It follows that F is a 2-extra vertex cut. Hence, we have κo (Gn ) ≤ 3n − 4.  4. The i-extra diagnosability of regular networks In this section, we will explore the i-extra diagnosability of regular networks Gn for 1 ≤ i ≤ 2 under PMC model and MM* model. For convenience, we use tiP (Gn ) (resp., tiM (Gn )) to denote the i-extra diagnosability of Gn under PMC model (resp., MM* model). 4.1. The 1-extra diagnosability of regular networks Firstly, we discuss the 1-extra diagnosability of regular networks under the PMC mdoel. In 2017, Lin et al.[23] investigated the relationship between i-extra connectivity and i-extra diagnosability of regular networks under the PMC model.

8

Lemma 11 [28] Given a general n-regular graph G and a integer h with 0 ≤ h ≤ n. Let κh (G) be the h-extra node connectivity of G and tph be the h-extra diagnosability of G under h of G the PMC model. If |V (G)| ≥ 2[κh (G) + h] + 1 and exists a connected subgraph S h )| = h + 1 and N (V (S h )) be the minimum h-extra-node-cut of G, then the such that |V (S relationship between h-extra connectivity and h-extra diagnosability of G under the PMC model is tph = κh (G) + h. Thus, according to Lemma 9 and Lemma 11, we obtain the 1-extra diagnosability of Gn under PMC model as follows. Theorem 12 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 4n − 2 and |E(i, j)| > 2n − 4 for n ≥ 3. Then, the 1-extra diagnosability of Gn under PMC model is t1P (Gn ) = 2n − 1. There is no such conclusion under the MM* model, so we cannot obtain results directly like PMC model. In the following, we start to explore the 1-extra diagnosability of regular networks under the MM* mdoel. Lemma 13 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 4n − 2 and |E(i, j)| > 2n − 4 for n ≥ 3. Then, the 1-extra diagnosability of Gn under the MM* model is t1M (Gn ) ≤ 2n − 1. Proof Let (u, v) be an edge of Gn and F1 = N ({u, v}), F2 = N [{u, v}]. Then, |F1 | = |N ({u, v})| = 2n − 2 and |F2 | = |N [{u, v}]| = 2n − 2 + 2 = 2n. Since F1 F2 = {u, v}, there is no edge between F1 F2 and V (Gn ) − F1 ∪ F2 . By Lemma 4, F1 and F2 are indistinguishable under the MM* model. Thus, Gn is not 1-extra  2n-diagnosable by Definition 2. Hence, we have t1M (Gn ) ≤ 2n − 1. Lemma 14 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 5n + 5 and |E(i, j)| > 2n − 4 for n ≥ 4. Then, the 1-extra diagnosability of Gn under MM* model is t1M (Gn ) ≥ 2n − 1. Proof Suppose, to the contrary, that t1M (Gn ) ≤ 2n−2. Then, there exist two distinct 1-extra faulty subsets F1 and F2 of Gn with |F1 | ≤ 2n − 1 and |F2 | ≤ 2n − 1, but the vertex set pair (F1 , F2 ) is not satisfied with any one condition in Lemma 4. Noting that |V (Gn )| > 5n + 5, V (Gn ) − F1 ∪ F2 = ∅. In the following, We will claim that V (Gn ) − F1 ∪ F2 has no isolated vertex. Suppose, by contradiction, there exists at least one isolated vertex in V (Gn ) − F1 ∪ F2 . Let W be the set of all isolated vertices in V (Gn ) − F1 ∪ F2 . Arbitrarily choose a vertex w ∈ W . We first show that F1 − F2 = ∅ and F2 − F1 = ∅. If F1 − F2 = ∅, then F1 ⊆ F2 . Noting that F2 is one 1-extra faulty set, w is an isolated vertex, a contradiction. Thus, F1 − F2 = ∅. Similarly, we have F2 − F1 = ∅. Next, we show that |NGn (w) ∩ (F1 − F2 )| = |NGn (w) ∩ (F2 − F1 )| = 1. If there exist two vertices, say u, v ∈ F1 − F2 , such that (u, w) ∈ E(Gn ) and (v, w) ∈ E(Gn ), then by Lemma 4, F1 and F2 are distinguishable, a contradiction. If there exists no vertex u ∈ F1 − F2 , such that (u, w) ∈ E(Gn ), then NGn (w) ⊆ F2 . Noting that F2 is one 1-extra faulty set, w is an isolated vertex, a contradiction. Thus, |NGn (w) ∩ (F1 − F2 )| = 1. Similarly, 9

we have |NGn (w) ∩ (F2 − F1 )| = 1. Then, for any w ∈ W , |NGn (w) ∩ (F1 ∩ F2 )| = n − 2. Thus, |F1 ∩ F2 | ≥ n − 2 and

|NGn [(F1 ∩F2 )∪W ] (w)| ≤ |W |(n − 2) w∈W

≤



dGn (v)

v∈F1 ∩F2

≤ |F1 ∩ F2 |n ≤ (|F2 | − 1)n ≤ (2n − 2)n. It follows that |W | ≤

(2n−2)n n−2

≤ 2n + 5. Let U = V (Gn ) − (F1 ∪ F2 ) − W . Then, we have

|U | = |V (Gn )| + |F1 ∩ F2 | − |F1 | − |F2 | − |W | > 5n + 5 + (n − 2) − 2(2n − 1) − (2n + 5) = 0, which implies that U = ∅. Considering that there exists no isolated vertex in U , there is no edge between U and F1 F2 by the assumption that F1 and F2 are indistinguishable. In addition, since W is the isolated vertex set of V (Gn ) − F1 ∪ F2 , there exists no edge between U and W . Thus, Gn − (F1 ∩ F2 ) is disconnected, which implies that F1 ∩ F2 is a vertex-cut. Moreover, noting that F1 and F2 are two distinct 1-extra faulty subsets, every component of Gn − F1 and Gn − F2 contains at least two vertices, and so every component of Gn − (F1 ∩ F2 ) contains at least two vertices. It follows that F1 ∩ F2 is one 1-extra vertex-cut. By Theorem 9, we have |F1 ∩ F2 | ≥ 2n − 2. Considering that |F1 | ≤ 2n − 1 and |F2 | ≤ 2n − 1, we have |F1 − F2 | = |F2 − F1 | = 1. Let F1 − F2 = {u} and F2 − F1 = {v}. Since |N (u) ∩ N (v)| ≤ 1, |W | = 1. Noting that there is no edge between U and (F1 F2 ) ∪ W , we have V (U ) ∩ N ({u, v, w}) = ∅ and so N ({u, v, w}) ⊆ F1 ∩F2 (see Fig. 2). Thus, we have |F1 ∩F2 | ≥ |N ({u, v, w})| = 3n−4. It follow that |F2 | = |F1 ∩ F2 | + |F2 − F1 | ≥ 3n − 4 + 1 = 3n − 3 > 2n − 1 ≥ |F2 |, a contradiction. Thus, for any vertex x ∈ Gn − (F1 ∪ F2 ), there exists at least one vertex, say y, such that (x, y) ∈ E(Gn ). Since F1 and F2 are indistinguishable, there exists no edge between F1 F2 and Gn − (F1 ∪ F2 ), which implies that F1 ∩ F2 is a vertex-cut. In addition, noting that F1 and F2 are two distinct 1-extra faulty sets, F1 ∩ F2 is a 1-extra vertex-cut. Thus, we have |F1 − F2 | ≥ 2 and |F1 ∩ F2 | ≥ 2n − 2 by Theorem 9. It follows |F1 | = |F1 − F2 | + |F1 ∩ F2 | ≥ 2 + 2n − 2 = 2n > 2n − 1 ≥ |F1 |, a contradiction. Hence, the lemma holds.



Combining Lemmas 13 and 14, we obtain the 1-extra diagnosability of Gn under the MM* model as follows. 10

w

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W

v

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U

Fig. 2: An illustration of the proof of Lemma 14. Theorem 15 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 5n + 5 and |E(i, j)| > 2n − 4 for n ≥ 4. Then, the 1-extra diagnosability of Gn under MM* model is t1M (Gn ) = 2n − 1. 4.2. The 2-extra diagnosability of regular networks We first establish the 2-extra diagnosability of regular networks under the PMC model by Lemma 10 and Lemma 11. Theorem 16 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 6n − 4 and |E(i, j)| > 2n − 4 for n ≥ 4. Then, the 2-extra diagnosability of Gn under PMC model is t2P (Gn ) = 3n − 2. Next, we will discuss the 2-extra diagnosability of regular networks under the MM* model. Lemma 17 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 6n − 4 and |E(i, j)| > 2n − 4 for n ≥ 4. Then, the 2-extra diagnosability of Gn under the MM* model is t2M ≤ 3n − 2. Proof Let u, v, w be a 2-path of Gn and F1 = N ({u, v, w}), F2 = N [{u, v, w}]. Then, we have |F1 | = |N ({u, v, w})| = n − 1 + n − 2 + n − 1 = 3n − 4 and |F2 | = |N [{u, v, w}]| = 3n − 4 + 3 = 3n − 1. Since F1 F2 = {u, v, w}, there is no edge between F1 F2 and V (Gn ) − F1 ∪ F2 . By Lemma 4, F1 and F2 are indistinguishable under the MM* model. Thus, Gn is not 2-extra (3n − 1)-diagnosable by Definition 2. Hence, we have t2M ≤ 3n − 2.  11

Lemma 18 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 8n + 4 and |E(i, j)| > 2n − 4 for n ≥ 4. Then, the 2-extra diagnosability of Gn under the MM* model is t2M (Gn ) ≥ 3n − 2. Proof Suppose, to the contrary, that t2M (Gn ) ≤ 3n−3. Then, there exist two distinct 2-extra faulty subsets F1 and F2 of Gn with |F1 | ≤ 3n − 2 and |F2 | ≤ 3n − 2, but the vertex set pair (F1 , F2 ) is not satisfied with any one condition in Lemma 4. Noting that |V (Gn )| > 8n + 4, we have V (Gn ) − F1 ∪ F2 = ∅. In the following, we will claim that V (Gn ) − F1 ∪ F2 has no isolated vertex. Suppose, by contradiction, there exists at least one isolated vertex in V (Gn ) − F1 ∪ F2 . Let W be the set of all isolated vertices in V (Gn ) − F1 ∪ F2 . Arbitrarily choose a vertex w ∈ W . Similar to Lemma 14, we have |NGn (w) ∩ (F1 − F2 )| = |NGn (w) ∩ (F2 − F1 )| = 1. Then, for any w ∈ W , |NGn (w) ∩ (F1 ∩ F2 )| = n − 2. Thus, |F1 ∩ F2 | ≥ n − 2 and

|NGn [(F1 ∩F2 )∪W ] (w)| ≤ |W |(n − 2) w∈W

≤



dGn (v)

v∈F1 ∩F2

≤ |F1 ∩ F2 |n ≤ (|F2 | − 1)n ≤ (3n − 3)n. It follows that |W | ≤

(3n−3)n n−2

≤ 3n + 6. Let U = V (Gn ) − (F1 ∪ F2 ) − W . Then, we have

|U | = |V (Gn )| + |F1 ∩ F2 | − |F1 | − |F2 | − |W | > 8n + 4 + (n − 2) − 2(3n − 2) − (3n + 6) = 0, which implies that U = ∅. Noting that there exists no isolated vertex in U , there is no edge between U and F1 F2 by the assumption that F1 and F2 are indistinguishable. Thus, F1 is a vertex-cut. Since F1 is a 2-extra fault-set of Gn , every component, say Ui , of U has |V (Ui )| ≥ 3 and every component, say Wi , of Gn [W ∪(F2 −F1 )] has |V (Wi )| ≥ 3. It follows that F1 is a 2-extra vertex-cut of Gn . Therefore, by Theorem 9, we have 3n − 4 ≤ |F1 | ≤ 3n − 2. Noting that every component Wi of Gn [W ∪ (F2 − F1 )] has |V (Wi )| ≥ 3, we have |F2 − F1 | ≥ 2. Similarly, we deduce that F2 is a 2-extra vertex-cut of Gn and |F1 − F2 | ≥ 2. In addition, since W is the isolated vertex set of V (Gn ) − F1 ∪ F2 , there is no edge between U and W . Thus, Gn − F1 ∩ F2 is disconnected, which implies that F1 ∩ F2 is a vertex-cut. Moreover, considering that F1 and F2 are two distinct 2-extra faulty subsets, every component of Gn − F1 and Gn − F2 contains at least three vertices, and so every component of Gn − (F1 ∩ F2 ) contains at least three vertices. It follows that F1 ∩ F2 is one 2-extra vertex-cut. By Theorem 9, we have |F1 ∩ F2 | ≥ 3n − 4. Noting that |F1 | ≤ 3n − 2 and |F2 | ≤ 3n − 2, we have |F1 − F2 | ≤ 2 and |F2 − F1 | ≤ 2. As a result, |F1 | = |F2 | = 3n − 2 and |F1 − F2 | = |F2 − F1 | = 2. Let F1 − F2 = {u1 , v1 } and F2 − F1 = {u2 , v2 }. 12

In the following, we assert that |W | ≤ 2. Suppose, to the contrary, that |W | ≥ 3. Let W = {w1 , w2 , w3 } and v1 , u1 , w1 , u2 , v2  be a path in Gn . Suppose that w2 is adjacent to u1 . If w2 is adjacent to u2 , then there exists a 4-cycle, a contradiction. If w2 is adjacent to v2 , then there exists a 5-cycle, a contradiction. Hence, w2 is adjacent to v1 . If w2 is adjacent to u2 , then there exists a 5-cycle, a contradiction. Thus, w2 is adjacent to v2 . Suppose that w3 is adjacent to u1 . If w3 is adjacent to u2 , then there exists a 4-cycle, a contradiction. If w3 is adjacent to v2 , then there exists a 5-cycle, a contradiction. Similarly, w3 cannot adjacent to v1 . In summary, |W | ≤ 2. Therefore, we have the following two cases. Case 1. |W | = 1. Without loss of generality, we suppose that w1 is exactly one isolated vertex in W . Noting that there is no edge between U and (F1 F2 )∪W , we have V (U )∩N ({v1 , u1 , u2 , v2 , w1 }) = ∅ and so N ({v1 , u1 , u2 , v2 , w1 }) ⊆ F1 ∩ F2 (see Fig. 3).

w

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v

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U Fig. 3: An illustration of the proof that W contains one isolated vertex of Lemma 18. Thus, we have |F1 ∩ F2 | ≥ |N ({v1 , u1 , u2 , v2 , w1 })| = 2(n − 1) + 3(n − 2) − 1 = 5n − 9, which implies |F2 | = |F1 ∩ F2 | + |F2 − F1 | ≥ 5n − 9 + 2 = 5n − 7 > 3n − 2 ≥ |F2 |, a contradiction. Case 2. |W | = 2. Without loss of generality, we suppose that w1 and w2 are isolated vertices in W . Noting that there is no edge between U and (F1 F2 )∪W , we have V (U )∩N ({v1 , u1 , u2 , v2 , w1 , w2 }) = ∅ and so N ({v1 , u1 , u2 , v2 , w1 , w2 }) ⊆ F1 ∩ F2 (see Fig. 4).

13

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v

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Fig. 4: An illustration of the proof that W contains two isolated vertices of Lemma 18. Thus, we have |F1 ∩ F2 | ≥ |N ({v1 , u1 , u2 , v2 , w1 , w2 })| = 6(n − 2). It follow that |F2 | = |F1 ∩ F2 | + |F2 − F1 | ≥ 6(n − 2) + 2 = 6n − 10 > 3n − 2 ≥ |F2 |, a contradiction. Thus, for any vertex x ∈ Gn − (F1 ∪ F2 ), there exists at least one vertex, say y, such that (x, y) ∈ E(Gn ). Since F1 and F2 are indistinguishable, there exists no edge between F1 F2 and Gn − (F1 ∪ F2 ), which implies that F1 ∩ F2 is a vertex-cut. In addition, noting that F1 and F2 are two distinct 2-extra faulty sets, F1 ∩ F2 is a 2-extra vertex-cut. Thus, we have |F1 − F2 | ≥ 3 and |F1 ∩ F2 | ≥ 3n − 4 by Theorem 9. It follows |F1 | = |F1 − F2 | + |F1 ∩ F2 | ≥ 3 + 3n − 4 = 3n − 1 > 3n − 2 ≥ |F1 |, a contradiction. Hence, the lemma holds.



Combining Lemmas 17 and 18, we obtain the 2-extra diagnosability of Gn under MM* model as follows. Theorem 19 Let Gn = (V (Gn ), E(Gn )) be an n-connected network with |V (Gn )| > 8n + 4 and |E(i, j)| > 2n − 4 for n ≥ 4. Then, the 2-extra diagnosability of Gn under the MM* model is t2M (Gn ) = 3n − 2.

14

4.3. i-extra diagnosis algorithm of regular networks In this section, our diagnostic schemes of Gn under two models are described. We first discuss the i-extra diagnosis algorithm of Gn under the PMC model. Definition 20 [36] Given a graph G = (V, E) and a syndrome σ on G produced by a faulty set. The 0-test subgraph of G, denoted T0 (G), is a subgraph of G defined by V (T0 (G)) ⊂ V and E(T0 (G)) = {(u, v) ∈ E | σ(u, v) = σ(v, u) = 0}. Lemma 21 [36] Given a graph G = (V, E) and a syndrome σ on G under the PMC model. (1) Let u, v be two adjacent nodes of G. If σ(u, v) = σ(v, u) = 0, then either both u and v are fault-free, or both u and v are faulty. (2) Let S be a connected component of T0 (G). Then either all nodes of S are fault-free, or all nodes of S are faulty. Algorithm 1 g-EP G-DIAG Input: An n-connected network Gn = (V (Gn ), E(Gn )) with |V (Gn )| > 6n − 4 for n ≥ 4; the syndrome σ on Gn under the PMC model; an integer t, the upper bound on the number of faulty vertices with t = (i + 1)n − 2i for i = 1, 2. Output: A fault-free vertex-set A and a faulty vertex-set B. 1. (Initialization) A = ∅, B = ∅, C = ∅ and D = V (Gn ). 2. Check all test results on Gn , and remove all edges with 1 ↔ 0, 0 ↔ 1, and 1 ↔ 1. Add all edges with 0 ↔ 0 to C and let T0 (Gn ) = Gn [C]. 3. Find the largest component S by Breadth-First Search in T0 (Gn ), and add all vertices in S to A. Let D = D − A. 4. For each undiagnosed vertex x in the previous steps, if x has a neighbor y such that y ∈ A, then add x to B according to σ(x, y) = 1. Let D = D − B. 5. For each undiagnosed vertex x in the previous steps, if |B| = (i + 1)n − i, then add all vertices in D into A. 6. return A fault-free vertex-set A and a faulty vertex-set B. Theorem 22 The time complexity of Algorithm 1 is O(nN ), where N is the number of vertices in Gn . Proof In step 2, there are two tests performed on each edge, thus the time complexity of this step is O( dv ) = O(2|E(Gn )|) = O(|E(Gn )|). Obviously, the time complexities v∈V (Gn )

of step 3, step 4 and step 5 are O(N ). Thus, the total time complexity of Algorithm 1 is  O(|E(Gn )|) + O(N ) = O(nN ) + O(N ) = O(nN ). Remark 1 Considering n < N , the time complexity of Algorithm 1 is O(nN ) ≤ O(N 2 ). Thus, it is a polynomial time algorithm on the vertex number of Gn . In the following, we start to explore the i-extra diagnosis algorithm of Gn under the MM* model.

15

Definition 23 Given a graph G = (V, E) and a syndrome σ on G produced by a faulty set. The 0-test subgraph of G, denoted T0 (G), is a subgraph of G defined by V (T0 (G)) ⊂ V and E(T0 (G)) = {(u, w) ∈ E, (v, w) ∈ E | σ(u, v; w) = 0}. According to the test rules, the following lemma can be easily obtained. Lemma 24 Given a graph G = (V, E) and a syndrome σ on G under the MM* model. Let S be a connected component of T0 (G). Then either all nodes of S are fault-free, or all nodes of S are faulty. Algorithm 2 g-EM G-DIAG Input: An n-connected network Gn = (V (Gn ), E(Gn )) with |V (Gn )| > 8n + 4 for n ≥ 4; the syndrome σ on Gn under the MM* model; an integer t, the upper bound on the number of faulty vertices with t = (i + 1)n − 2i for i = 1, 2. Output: A fault-free vertex-set A and a faulty vertex-set B. 1. (Initialization) A = ∅, B = ∅, C = ∅ and D = V (Gn ). 2. Check all test results on Gn , and remove all edges with σ(u, v; w) = 1. Add all edges with σ(u, v; w) = 0 to C and let T0 (Gn ) = Gn [C]. 3. Find the largest component S by Breadth-First Search in T0 (Gn ), and add all vertices in S to A. Let D = D − A. 4. For each undiagnosed vertex x in the previous steps, if x has a neighbor y such that y ∈ A, and y has a neighbor z such that z ∈ A, then add x to B according to σ(x, z; y) = 1. Let D = D − B. 5. For each undiagnosed vertex x in the previous steps, if |B| = (i + 1)n − i, then add all vertices in D into A. 6. return A fault-free vertex-set A and a faulty vertex-set B. Theorem 25 The time complexity of Algorithm 2 is O(n2 N ), where N is the number of vertices in Gn . Proof In step 2, there is a test performed on two edges which are shared by a common vertex, thus the time complexity is O( n(n−1) N ). In addition, the time complexities of step 3, step 4 2 N )+O(N ) = and step 5 are O(N ). Hence, the total time complexity of Algorithm 2 is O( n(n−1) 2 2 O(n N ).  Remark 2 Considering n < N , the time complexity of Algorithm 2 is O(n2 N ) ≤ O(N 3 ). Thus, it is a polynomial time algorithm on the vertex number of Gn . 5. Applications to some networks In this section, we derive the {1, 2}-extra connectivities and {1, 2}-extra diagnosabilities of some networks by the results in Section 3 and Section 4.

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5.1. Star networks Sn Definition 26 [1] An n-dimensional star graph Sn (see Fig. 5) is defined to be a graph G = G(V, E) such that (1) the vertex set V is the set of all permutations of n; (2) the edge set E = {(x, y) | x, y ∈ V, x = x1 x2 · · · xi−1 xi xi+1 · · · xn , y = xi x2 · · · xi−1 x1 xi+1 · · · xn , 2 ≤ i ≤ n}. 1234 12

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Fig. 5: The star networks S2 , S3 and S4 . Proposition 27 [1, 3, 19] Sn has the following properties: edges. (1) Sn is (n − 1)-regular with n! nodes and (n−1)n! 2 (2) κ(Sn ) = n − 1. (3) g(Sn ) = 6 for n ≥ 3. (4) Sn can be decomposed into n vertex-disjoint subgraphs, denoted Sni , by fixing the symbol in the last position n, in which the symbol in the nth position is i, where 1 ≤ i ≤ n. Obviously, Sni is isomorphic to Sn−1 . (5) For any i = j ∈ n, the number of cross edges between Sni and Snj is (n − 2)!. Theorem 28 The {1, 2}-extra connectivity and {1, 2}-extra diagnosabilities of the star graph Sn (n ≥ 6) are listed as follows. (1) (2) (1) κo (Sn ) = 2n − 4 and κo (Sn ) = 3n − 7. (2) t1P (Sn ) = 2n − 3 and t1M (Sn ) = 2n − 3. (3) t2P (Sn ) = 3n − 5 and t2M (Sn ) = 3n − 5. Proof Obviously, the star graph Sn satisfies the conditions of Gn . According to Theorem 9 (1) (2) and Theorem 10, the {1, 2}-extra connectivities of Sn are κo (Sn ) = 2n − 4 and κo (Sn ) = 17

3n − 7. By Theorem 12 and Theorem15, the 1-extra diagnosabilities of Sn under the PMC model and MM* model are t1P (Sn ) = 2n − 3 and t1M (Sn ) = 2n − 3. Moreover, Through Theorem 16 and Theorem 19, the 2-extra diagnosabilities of Sn under PMC model and MM* model are t2P (Sn ) = 3n − 5 and t2M (Sn ) = 3n − 5.  5.2. Pancake networks Pn Definition 29 [1] An n-dimensional pancake graph Pn (see Fig. 6) is defined to be a graph G = (V, E) such that (1) the node set V is the set of all permutations of n; (2) the edge set E = {(u, (u)i ) | u, (u)i ∈ V, u = x1 x2 · · · xi · · · xn , (u)i = xi xi−1 · · · x2 x1 xi+1 · · · xn , 2 ≤ i ≤ n}, where (u)i denotes the unique i-neighbor of u for 2 ≤ i ≤ n.

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Fig. 6: The pancake networks P2 , P3 and P4 . Proposition 30 [1, 6, 16] The n-dimensional pancake graph Pn (n ≥ 2) has the following properties: (1) Pn is (n − 1)-regular with n! vertices and (n−1)n! edges; 2 (2) κ(Pn ) = n − 1; (3) g(Pn ) = 6 for n ≥ 3; (4) Pn can be decomposed into n vertex-disjoint subgraphs, denoted Pni , by fixing the symbol in the last position n, in which the symbol in the nth position is i, where 1 ≤ i ≤ n. Obviously, Pni is isomorphic to Pn−1 . (5) For any i = j ∈ n, the number of cross edges between Pni and Pnj is (n − 2)!. Theorem 31 The {1, 2}-extra connectivity and {1, 2}-extra diagnosabilities of the pancake graph Pn (n ≥ 6) are listed as follows. (1) (2) (1) κo (Pn ) = 2n − 4 and κo (Pn ) = 3n − 7. (2) t1P (Pn ) = 2n − 3 and t1M (Pn ) = 2n − 3. (3) t2P (Pn ) = 3n − 5 and t2M (Pn ) = 3n − 5. 18

Proof Obviously, the pancake graph Pn satisfies the conditions of Gn . According to Theorem (1) (2) 9 and Theorem 10, the {1, 2}-extra connectivities of Pn are κo (Pn ) = 2n − 4 and κo (Pn ) = 3n − 7. By Theorem 12 and Theorem15, the 1-extra diagnosabilities of Pn under the PMC model and MM* model are t1P (Pn ) = 2n − 3 and t1M (Pn ) = 2n − 3. Moreover, Through Theorem 16 and Theorem 19, the 2-extra diagnosabilities of Pn under PMC model and MM* model are t2P (Pn ) = 3n − 5 and t2M (Pn ) = 3n − 5.  5.3. Burnt pancake networks BPn Let n be a positive integer. To save space, the negative sign may be placed on the top of an expression. Thus, ¯i = −i. We use [n] to denote the set n ∪ {¯i | i ∈ n}. A signed permutation of n is an n-permutation u1 u2 · · · un of [n] such that |u1 ||u2 | · · · |un |, taking the absolute value of each element, forms a permutation of n. Definition 32 [5] An n-dimensional burnt pancake graph BPn (see Fig. 7) is defined to be a graph G = (V, E) such that (1) the node set V is the set of all the signed permutations of n; (2) the edge set E = {(u, (u)i ) | u, (u)i ∈ V, u = x1 x2 · · · xi · · · xn , (u)i = xi xi−1 · · · x2 x1 xi+1 · · · xn , 1 ≤ i ≤ n}, where (u)i denotes the unique i-neighbor of u for 1 ≤ i ≤ n. 321 321

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Fig. 7: The burnt pancak networks BP1 , BP2 and BP3 .

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Proposition 33 [5, 15, 30] The n-dimensional pancake graph BPn (n ≥ 2) has the following properties: (1) BPn is n-regular with n! × 2n vertices and n × n! × 2n−1 edges; (2) κ(BPn ) = n; (3) g(BPn ) = 8 for n ≥ 2; (4) BPn can be decomposed into 2n vertex-disjoint subgraphs, denoted BPni , by fixing the symbol in the last position n, in which the symbol in the nth position is i, where 1 ≤ i ≤ n. Obviously, BPni is isomorphic to BPn−1 ; (5) For any i = j ∈ [n], the number of cross edges between BPni and BPnj is (n−2)!×2n−2 . Theorem 34 The {1, 2}-extra connectivity and {1, 2}-extra diagnosabilities of the burnt pancake graph BPn (n ≥ 4) are listed as follows. (1) (2) (1) κo (BPn ) = 2n − 2 and κo (BPn ) = 3n − 4. (2) t1P (BPn ) = 2n − 1 and t1M (BPn ) = 2n − 1. (3) t2P (BPn ) = 3n − 2 and t2M (BPn ) = 3n − 2. Proof Obviously, the burnt pancake graph BPn satisfies the conditions of Gn . According to (1) Theorem 9 and Theorem 10, the {1, 2}-extra connectivities of BPn are κo (BPn ) = 2n − 2 (2) and κo (BPn ) = 3n − 4. By Theorem 12 and Theorem15, the 1-extra diagnosabilities of BPn under the PMC model and MM* model are t1P (BPn ) = 2n − 1 and t1M (BPn ) = 2n − 1. Moreover, Through Theorem 16 and Theorem 19, the 2-extra diagnosabilities of BPn under PMC model and MM* model are t2P (BPn ) = 3n − 2 and t2M (BPn ) = 3n − 2.  Remark 3 The {1, 2}-extra connectivities and {1, 2}-extra diagnosabilities of Sn , Pn and BPn under the PMC model have been obtained in the literature, but the {1, 2}-extra diagnosabilities of them under the MM* model are still open. According to our results, the {1, 2}-extra diagnosabilities of them under the MM* model can be directly obtained. 6. Conclusion The extra connectivity and the extra diagnosability are two excellent indicators to evaluate the reliability of the multiprocessor system and provide many promising applications to complex networks and data center networks. The paper is devoted to establishing the i-extra connectivity and the i-extra diagnosability of general regular networks with some conditions for 1 ≤ i ≤ 2. We prove that the i-extra connectivities of regular networks with some conditions Gn are (i + 1)n − 2i for i = 1, 2, where n ≥ 4, and the i-extra diagnosabilities of the networks under the PMC model and MM* model are (i + 1)n − i for i = 1, 2, where n ≥ 4. Furthermore, by our results, the i-extra connectivities and i-extra diagnosabilities under the PMC model and MM* model of some networks can be obtained directly. However, investigating these two metric parameters with larger i (≥ 3) is meaningful and challenging. Acknowledgment This work was supported by the National Natural Science Foundation of China (No. U1905211, No. 61572337, No. 61602333, and No. 61972272), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJA520009), and the Priority Academic Program Development of Jiangsu Higher Education Institutions. 20

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Conflict of Interest Statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “The extra connectivity and extra diagnosability of regular interconnection networks”.

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