Fault diagnosability of regular graphs

An interconnection network’s diagnosability is an important measure of its selfdiagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the h-good-neighbor conditional diagnosability, which requires that every fault-free node has at least h fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The h-goodneighbor diagnosability under the PMC (resp. MM*) model of a graph G, denoted by tPMC h (G) (resp. t MM∗ h (G)), is the maximum value of t such that G is h-good-neighbor t-diagnosable under the PMC (resp. MM*) model. In this paper, we study the 2-good-neighbor diagnosability of some general k-regular kconnected graphs G under the PMC model and the MM* model. The main result tPMC 2 (G) = t MM∗ 2 (G) = g(k − 1)− 1 with some acceptable conditions is obtained, where g is the girth of G. Furthermore, the following new results under the two models are obtained: tPMC 2 (HSn) = t MM∗ 2 (HSn) = 4n− 5 for the hierarchical star network HSn, t PMC 2 (S 2 n) = t MM∗ 2 (S 2 n) = 6n− 13 for the split-star networks S2 n and tPMC 2 (Γn(∆)) = t MM∗ 2 (Γn(∆)) = 6n − 16 for the Cayley graph generated by the 2-tree Γn(∆).


Introduction
A multiprocessor system is modeled as an undirected simple graph G = (V, E), whose vertices (nodes) represent processors and edges (links) represent communication links.
With the rapid development of multiprocessor systems, processor failure is inevitable along with the number of processors increasing. The process of identifying all the faulty units in a system is called system-level diagnosis. For the purpose of self-diagnosis of a system, a number of models have been proposed for diagnosing faulty processors in a network. Among the proposed models, PMC model (that is, Preparata, Metze and Chien's model) [17] and comparison model (MM* model) [16] are widely used. In the PMC model, the diagnosis of the system is achieved through two linked processors testing each other. In the MM* model, to diagnose the system, a processor sends the same task to two of its neighbors, and then compares their responses. The PMC and MM* models have been extensively investigated.
A system is said to be t-diagnosable if all faulty units can be identified provided the number of faulty units present does not exceed t. The diagnosability is the maximum number of faulty processors which can be correctly identified. The classical diagnosability of a network is quite small owing to the fact that it ignores the unlikelihood of some specific processors failing at the same time. In 2005, Lai et al. [13] introduced a restricted diagnosability of the system called conditional diagnosability by assuming that it is almost impossible that all neighbors of one vertex are faulty simultaneously. Inspired by this concept, Peng et al. [18] then proposed the h-good-neighbor diagnosability, which requires every fault-free vertex has at least h fault-free neighbors. Furthermore, they evaluated the h-good-neighbor diagnosability of the n-dimensional hypercube Q n under the PMC model. Yuan et al. [24] and [25] studied the h-good-neighbor diagnosability of the kary n-cubes (k ≥ 4) and 3-ary n-cubes, respectively, under the PMC model and MM* model. Wang et al. [20] and [21] determined the 2-good-neighbor diagnosability of the Cayley graph generated by transposition trees Γ n and the alternating graph network AN n , respectively. More results can be found in [14,15] etc.
In this paper, we study the 2-good-neighbor diagnosability of some general k-regular k-connected graphs G under the PMC model and the MM* model and obtain the relationship between 2-good-neighbor diagnosability and R 2 -connectivity κ 2 (G). The main result t P M C 2 (G) = t M M * 2 (G) = κ 2 (G) + g − 1 = g(k − 1) − 1 under the two models with some acceptable conditions is obtained, where g is the girth of G. More precisely, our main result is the following Theorem 1.
Theorem 1. Let G be a k-regular k-connected graph of order N (that is, with N vertices). Let g be the girth of G, = cn(G) be the maximum number of common neighbors between any two vertices and = ca(G) be the maximum number of common neighbors between any two adjacent vertices. Suppose further that all of the following conditions hold: (2) ≤ 2 and ≤ 1, and Then, if g = 3 and G contains no 5-cycles; 3 if g = 3 and G contains 5-cycles; 2 if g = 4 and G contains no 5-cycles; 4 if g = 4 and G contains 5-cycles; Furthermore, the following new results about the 2-good-neighbor diagnosability t 2 (G) under the PMC model and MM* model are obtained: (Γ n (∆)) = 6n−16 for Cayley graph generated by the 2-tree Γ n (∆). Especially, the relationship t P M C (G)) and κ 2 (G) are given. In the literature, most known results about 2-goodneighbor conditional diagnosability of some networks are gotten independently, and some proofs are longwinded. As consequences of our results, some of them can be obtained easily.
The remainder of this paper is organized as follows. Section 2 introduces necessary definitions. Our main results are given in Section 3. As applications of our main results, Section 4 concentrates on the applications to some popular interconnection networks. Finally, our conclusions are given in Section 5.
2 Throughout this paper, all graphs are finite, undirected and without loops. We follow [22] for terminologies and notations not defined here.
Let G = (V (G), E(G)) be a graph. For a vertex u ∈ V (G), we use the symbol N G (u) to denote a set of vertices in G adjacent to u. The cardinality |N G (u)| represents the degree of u in G, denoted by d G (u), and δ(G) is the minimum degree of G. For a vertex The components of G are its maximal connected subgraphs. The connectivity κ(G) of a connected graph G is the minimum number of vertices to be removed from G so that the resulting graph is either disconnected or trivial.
For two adjacent vertices u and v in G, let cn(G; u, v) denote the number of vertices who are the neighbors of both u and v, that is, For a positive integer n, let [n] = {1, 2, . . . , n}. For a finite group A and a subset S of A such that 1 / ∈ S (where 1 is the identity element of A) S = S −1 (that is, s ∈ S implies s −1 ∈ S), the Cayley graph Cay(A; S) on A with respect to S is defined to have vertex set A and edge set {(g, gs)|g ∈ A, s ∈ S}. (Generally, S = S −1 is not required in the definition of a Cayley graph. We impose the condition here so that the corresponding Cayley graph can be treated as undirected.) An h-good-neighbor cut of a graph G is an h-good-neighbor faulty set F such that G−F is disconnected. The minimum cardinality of h-good-neighbor cuts is said to be the R h -connectivity (or h-good-neighbor connectivity) of G, denoted by κ h (G). The parameter κ 1 (G) is equal to extra connectivity κ 1 (G) proposed by Fábrega and Fiol [8], where κ k (G) is the cardinality of a minimum set S ⊆ V (G) such that G − S is disconnected and each component of G − S has at least k + 1 vertices. The symmetric difference of F 1 ⊆ V (G) and F 2 ⊆ V (G) is defined as the set The following two lemmas which characterize a graph for h-good-neighbor t-diagnosable under the PMC model and the MM* model, respectively. These lemmas essentially turn the diagnosability problem into a graph theory problem.
and v ∈ F 1 ∆F 2 for each distinct pair of h-good-neighbor faulty sets F 1 and F 2 of V with |F 1 | ≤ t and |F 2 | ≤ t.
The h-good-neighbor diagnosability under the PMC model of a graph G, denoted by t P M C h (G), is the maximum value of t such that G is h-good-neighbor t-diagnosable under the PMC model.

3
(1) There are two vertices u, w ∈ V \ (F 1 ∪ F 2 ) and there is a vertex v ∈ F 1 ∆F 2 such that (u, v) ∈ E and (u, w) ∈ E.
(2) There are two vertices u, v ∈ F 1 \ F 2 and there is a vertex The h-good-neighbor diagnosability under the MM* model of a graph G, denoted by

Main result
In this section, we will determine the 2-good-neighbor diagnosability of some general kregular k-connected graphs G under the PMC model and the MM* model. Before we prove Theorem 1, we would like to comment that a cycle is the most basic connected graph with minimum degree 2. Thus, to find a minimum 2-good-neighbor faulty set, it is natural to find a small cycle in the graph and delete its neighbors. Since the graph is k-regular, one would expect to delete "g(k − 2)" vertices. However, this assumes that all these g(k − 2) neighbors are distinct. This is addressed by the conditions on cn and ca. The conditions N ≥ 2g(k − 1) − 1 and k ≥ ξ + 2 are technical. If the girth is not large enough, there are additional difficulties. Thus the condition on k is much simpler if g ≥ 9. In fact, if g ≥ 9, the requirement reduces to k ≥ 3, which is mild as this excludes only cycles.
Proof of Theorem 1. Let C = (v 1 , v 2 , . . . , v g , v 1 ) be a shortest cycle in G. The following two claims are useful.
Proof of Claim 1. Obviously, (We remark that the declaration is clearly true if g is sufficiently large; otherwise, this creates a smaller cycle. For graphs with smaller girth, the argument is more technical.) We consider the claim according to the girth of G as follows.
If g = 7, then by Condition (2), If g ≥ 9, then by Condition (2), By the above discussion, for any . This establishes Claim 2.
(I) First, we consider t P M C 2 (G).
By Claim 1, S 1 and S 2 are both 2-good-neighbor faulty sets of G with |S 1 | = g(k − 2) and Proof of Claim 3. By Lemma 1, it is equivalent to prove: For each distinct pair of 2good-neighbor faulty sets F 1 and Suppose, on the contrary, that there are two distinct 2-good-neighbor faulty sets F 1 and Without loss of generality, assume that then the claim is clearly true. Henceforth, we may assume that F 1 ∩ F 2 = ∅. Note that since F 1 is a 2-good-neighbor faulty set, δ(G−F 1 ) ≥ 2. Similarly, since F 2 is a 2-good-neighbor faulty set, δ(G−F 2 ) ≥ 2.

5
Because there are no edges between V (G) \ (F 1 ∪ F 2 ) and has a cycle, say C 1 , and the length of C 1 is at least g as the girth of G is g. It now follows that Thus Claim 3 holds.
(II) Now we consider t M M * 2 (G).
We first prove t M M * 2 (G) ≤ g(k−1)−1. By Claim 5, S 1 and S 2 are both 2-good-neighbor faulty sets of G with |S 1 | = g(k − 2) and ) ∩ V (C) = ∅, and S 1 and S 2 do not satisfy any condition in Lemma 2, so G is not 2-good-neighbor g(k − 1)-diagnosable. Thus In the following we prove Suppose, on the contrary that there are two distinct 2-goodneighbor faulty sets F 1 and F 2 of G with |F 1 | ≤ g(k − 1) − 1 and |F 2 | ≤ g(k − 1) − 1, but (F 1 , F 2 ) does not satisfy any one of the conditions in Lemma 2. Clearly, Proof of Claim 4. Since F 1 is a 2-good-neighbor faulty set, |N G−F 1 (x)| ≥ 2 for any x ∈ V (G)\F 1 . Note that the vertex set pair (F 1 , F 2 ) does not satisfy any one of the conditions in Lemma 2, by the Condition (3) of Lemma 2, for any pair of vertices u, v ∈ F 2 \F 1 , there is no vertex w ∈ V (G) \ (F 1 ∪ F 2 ) such that (u, w) ∈ E(G) and (v, w) ∈ E(G). Thus, any vertex x ∈ V (G)\(F 1 ∪F 2 ) has at most one neighbor in F 2 \F 1 , |N G−(F 1 ∪F 2 ) (x)| ≥ 2−1 = 1, this implies every vertex of G − (F 1 ∪ F 2 ) is not an isolated vertex. The proof of Claim 4 is finished.
If F 1 ∩ F 2 = ∅, then the claim is clearly true. Henceforth, we may assume that . By Claim 4, y has at least one neighbor in G − (F 1 ∪ F 2 ). Note that the vertex set pair (F 1 , F 2 ) does not satisfy any one of the conditions in Lemma 2, by Condition (3) of Lemma 2, y has no neighbor in F 1 ∆F 2 . Since y is arbitrary, there are no edges between V (G) \ (F 1 ∪ F 2 ) and F 1 ∆F 2 .
Since F 2 \F 1 = ∅, and F 1 is a 2-good-neighbor faulty set, by Condition (3) has a cycle C 1 with length at least g as the girth of G is g, it follows that |F 2 \ F 1 | ≥ g. Then, By Theorem 1 and Claim 2, the following Theorem 2 is obtained.
6 Theorem 2. Let G be a k-regular and k-connected graph and g be the girth of G. If G satisfies all the conditions in Theorem 1, then t P M C 2 (G) = t M M * 2 (G) = κ 2 (G) + g − 1 = g(k − 1) − 1.

Applications to some networks
As applications of Theorem 1 and Theorem 2, in this section, we determine the 2-goodneighbor diagnosability and the R 2 -connectivity for some networks.

Definition 2. ( [19])
An n-dimensional hierarchical star network HS(n, n), or simply HS n , is made of n! n-dimensional star graphs S n , called modules. Each node of HS n is denoted by a two-tuple address (x, y), where both x and y are arbitrary permutations of n distinct symbols. The first n-bit permutation x identifies the module of x and the second n-bit permutation y identifies the position of y inside its module. Two nodes (x, y) and (x , y ) in HS n are adjacent, if one of the following three conditions holds: (1) x = x and (y, y ) ∈ E(S n ); That is, (x, y) is adjacent to (x, y ) if (y, y ) ∈ E(S n ).
(3) x = x , x = y and x = y , y = x . That is, (x, y) is adjacent to (y, x) if x = y.
The 3-dimensional hierarchical star HS 3 is shown in Fig. 1.

Remark 1.
Each node in HS n is assigned a label (x, y) = (x 1 x 2 · · · x n , y 1 y 2 · · · y n ), where x 1 x 2 · · · x n and y 1 y 2 · · · y n are permutations of n distinct symbols (not necessarily distinct from each other). The edges of the HS n are defined by the following n generators: where x(1, n) is the permutation by interchanging the nth element with 1st element of x.
Let (x, y) be a vertex of HS n . The neighbor set of (x, y) is exactly {h i ((x, y))|i ∈ I n }. Furthermore, h 1 ((x, y)) is called the extra neighbor of (x, y) and h i ((x, y)) is called the internal neighbor of (x, y) for 2 ≤ i ≤ n. Define HS x n to be an induced subgraph by the vertex set {(x, y) ∈ V (HS n ) : y ∈ V (S n )}, which is isomorphic to an n-dimensional star graph S n identified by x.

Remark 2.
Any vertex has exactly one extra neighbor in HS n , i.e., every vertex (x, y) in HS x n is exactly incident to one crossing edge (x, h 1 ((x, y))). There is one or two crossing edges between any pair of modules. Moreover, for a fixed module HS x n , there are two cross edges between HS x n and HS x(1,n) n ; there is only one cross edge between HS x n and HS y n , where y ∈ Γ n \ {x, y}.

Lemma 3. ( [19])
For any integer n ≥ 3, HS n is an n-regular n-connected graph, and its girth is 4. Any two vertices have at most two common neighbors in HS n .
Recall that HS n consists of n! modules, each module is isomorphic to the star graph S n , the known (fault tolerance) properties of S n are useful.
Lemma 4. Let U be a subset with 2 ≤ |U | ≤ 4 of n-dimensional star graph S n for n ≥ 5.
The following statements hold.
Lemma 5. Let F be a faulty subset of n-dimensional star graph S n for n ≥ 5. The following statements hold.
(1) ( [23]) If |F | ≤ 2n − 4, then S n − F is connected; or contains two components, one of which is an isolated vertex; or contains two components, one of which is an edge; furthermore, F is the neighborhood of this isolated edge with |F | = 2n − 4.
(2) ( [28]) If |F | ≤ 3n − 8, then S n − F is connected; or contains a large component and the union of smaller components which contain at most two vertices in total.
(3) ( [28]) If |F | ≤ 4n − 11, then S n − F is connected; or contains a large component and the union of smaller components which contain at most three vertices in total.

8
In the following, let F be a faulty subset of n-dimensional hierarchical star network HS n . For each α ∈ Γ n , let F α = F ∩ V (HS α n ) and f α = |F α |. Let I = {α : α ∈ Γ n and HS α n − F α is disconnected}, F I = α∈I F α , f I = |F I |, I = Γ n \ I, F I = α∈I F α , f I = |F I | and HS I n = HS n [ α∈I V (HS α n )]. These notations will be used throughout the paper. The following Claim holds.
Lemma 6. ( [9]) Let F be a faulty subset of V (HS n ) for n ≥ 5. If |F | ≤ 2n − 3, then HS n − F either is connected; or contains two components, one of which is an isolated vertex.
Lemma 7. Let F be a faulty subset of V (HS n ) for n ≥ 5. If |F | ≤ 3n − 6, then HS n − F either is connected; or contains a large component and the union of smaller components which contain at most two vertices in total.
In this case, I = Γ n , HS n − F = HS I n − F I is connected. Case 2. |I| = 1.
Without loss of generality, let I = {α}. We consider the following three cases.
Since HS α n is isomorphic to S n , by Lemma 5 (2), HS α n −F α contains a large component, say B, and the union of smaller components which contain at most two vertices in total. Since (n! − 1) − (3n − 6) − 2 − |I| = n! − 3n + 2 > 0 for n ≥ 5, by Claim 5, B is connected to HS I n − F I . Thus, HS n − F either is connected; or contains a large component and the union of smaller components which contain at most two vertices in total.
Since |F | ≤ 3n − 6, f I ≤ 1, by Remark 2, at most one vertex is disconnected with HS I n − F I in HS n − F . Thus, HS n − F either is connected; or contains two components, one of which is an isolated vertex.

Application to the Split-star network S 2 n
Cheng et al. [4] proposed the Split-star networks as alternatives to the star graphs and companion graphs with the alternating group graphs. Let S 2 n,E be a subgraph of S 2 n induced by the set of even permutations, in which the adjacency rule is precisely the 3-rotation. We know that S 2 n,E is the alternating group graph AG n . Let S 2 n,O be a subgraph of S 2 n induced by the set of odd permutations, in which the adjacency rule is precisely the 3-rotation. We have that S 2 n,O is also isomorphic to AG n and S 2 n,O is isomorphic to S 2 n,E via the 2-exchange φ(a 1 a 2 a 3 · · · a n ) = a 2 a 1 a 3 · · · a n . Hence, there are n! 2 independent edges between S 2 n,O and S 2 n,E .
Lemma 9. Let S 2 n be the n-dimensional split-star network.
Lemma 10. Let F be a vertex cut of AG n for n ≥ 5.
4.3 Application to the Cayley graph generated by 2-tree Γ n (∆) Definition 4. Let Γ be the alternating group, the set of even permutations on {1, 2, . . . , n}, and the generating set ∆ be a set of 3-cycles. The corresponding Cayley graph Cay(Γ, ∆) is denoted by Γ n (∆). To get an undirected Cayley graph, we will assume that whenever a 3-cycle (abc) is in ∆, so is its inverse (acb). We can depict ∆ via a graph H with vertex set [n], where a triangle K 3 on vertices a, b, and c corresponds to each pair of a 3-cycle (abc) and its inverse in ∆, where a hyperedge of size 3 corresponds to each pair of a 3-cycle and its inverse in ∆. We consider a simpler case when H has a tree-like structure. Such a graph is built by the following procedure. We start from K 3 , then repeatedly add a new vertex, joining it to exactly two adjacent vertices of the previous graph. Any graph obtained by this procedure is called a 2-tree. If v is a vertex of a 2-tree H with the property that H can be generated in such a way that v is the last vertex added, then v is called a leaf of the 2-tree. If ∆ is the set of 3-cycles via a 2-tree H, then Γ n (∆) is called the Cayley graph generated by 2-trees ∆.
The alternating group graph AG n [11], can be viewed as the Cayley graph generated by the graph having a tree-like (in fact, star-like) structure of triangles.  (2) Γ n (∆) does not contain K 4 − e, that is, K 4 with an edge deleted, and K 2,3 as a subgraph. For any two vertices u and v, cn(G : u, v) = 1 if u and v are adjacent, cn(G : u, v) ≤ 2 otherwise.
The 2-good neighbor diagnosability of the Cayley graph generated by the 2-tree Γ n (∆) has not been determined so far. By Theorem 2, we immediately get κ 2 (Γ n (∆)).

Concluding remarks
In this paper, the 2-good-neighbor diagnosability of a general regular graph G is studied. The main result t P M C 2 (G) = t M M * 2 (G) = κ 2 (G)+g−1 = g(k−1)−1 under some conditions is obtained. As consequences of our results, the 2-good-neighbor diagnosability and R 2connectivity κ 2 (G) of many networks including some known results can be obtained. The following new results are obtained: t P M C 2 (HS n ) = t M M * 2 (HS n ) = 4n − 5 for hierarchical star network HS n , t P M C 2 (S 2 n ) = t M M * 2 (S 2 n ) = 6n − 13 for split-star networks S 2 n , and t P M C 2 (Γ n (∆)) = t M M * 2 (Γ n (∆)) = 6n − 16 for Cayley graph generated by the 2-tree Γ n (∆).
In the literature, most known results about 2-good-neighbor conditional diagnosability of some networks are obtained via ad-hoc methods under various techniques. In this paper, we unified these approaches to obtain general results. As consequences of these results, some of them can be obtained easily.
Observing that Wang et al. [20] and [21] obtained 2-good-neighbor diagnosability of the Cayley graphs generated by transposition trees and the alternating group networks, respectively, under the PMC model and MM* model. We can deduce these results by Theorem 1 and Theorem 2 as directive corollaries. The details are omitted.
Furthermore, the h-good-neighbor diagnosability of a general regular graph G for h ≥ 3 is a challenging work which need to be studied in the future. Since we have established a relationship between the R 2 -connectivity and the 2-good-neighbor diagnosability of regular graphs (under certain conditions), any R 2 -connectivity result (even an upper bound result) for such an interconnection network will "automatically" give a 2-goodneighbor diagnosability result (respectively an upper bound result) for this network by Theorem 2. This advances the study of 2-good-neighbor diagnosability as one can now leverage on such existing results rather than applying ad-hoc methods. It would also be interesting to see whether we can apply Theorem 2 in "reverse," that is, find a network in the literature where its 2-good-neighbor diagnosability was obtained via an ad-hoc method but its R 2 -connectivity has not been evaluated.