The 2-Extra Diagnosability of Alternating Group Graphs under the PMC Model and MM * Model

Diagnosability of a multiprocessor system is one important study topic. In 2015, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, g-extra diagnosability, which restrains that every fault-free component has at least ( ) 1 g + fault-free nodes. As a favorable topology structure of interconnection networks, the n-dimensional alternating group graph AGn has many good properties. In this paper, we give that the 2-extra diagnosability of AGn is 6 17 n − for 5 n ≥ under the PMC model and MM* model.


Introduction
Many multiprocessor systems take interconnection networks (networks for short) as underlying topologies and a network is usually represented by a graph where nodes represent processors and links represent communication links between processors.We use graphs and networks interchangeably.For a multiprocessor system, study on the topological properties of its network is important.Furthermore, some processors may fail in the system, so processor fault identification plays an important role for reliable computing.The first step to deal with faults is to identify the faulty processors from the fault-free ones.The identification process is called the diagnosis of the system.A system is said to be t-diagnosable if all faulty processors can be identified without replacement, provided that the number of faults presented does not exceed t.The diagnosability of a system G is the maximum value of t such that G is t-diagnosable [1] [2] [3].For a t-diagnosable system, Dahbura and Masson [1] proposed an algorithm with time complex ( )

O n
, which can effectively identify the set of faulty processors.Several diagnosis models were proposed to identify the faulty processors.One major approach is the Preparata, Metze, and Chien's (PMC) diagnosis model introduced by Preparata et al. [4].The diagnosis of the system is achieved through two linked processors testing each other.Another major approach, namely the comparison diagnosis model (MM model), was proposed by Maeng and Malek [5].In the MM model, to diagnose a system, a node sends the same task to two of its neighbors, and then compares their responses.In 2005, Lai et al. [3] introduced a restricted diagnosability of multiprocessor systems called conditional diagnosability.They consider the situation that any fault set cannot contain all the neighbors of any vertex in a system.In 2012, Peng et al. [6] proposed a measure for fault diagnosis of the system, namely, g-good-neighbor diagnosability (which is also called g-good-neighbor conditional diagnosability), which requires that every fault-free node has at least g fault-free neighbors.In [6], they studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the PMC model.In [7], Wang and Han studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the MM* model.Yuan et al. [8] and [9] studied that the g-good-neighbor diagnosability of the k-ary n-cube ( ) under the PMC model and MM* model.The Cayley graph n CΓ gen- erated by the transposition tree n Γ has recently received considerable atten- tion.In [10] [11], Wang et al. studied the g-good-neighbor diagnosability of n CΓ under the PMC model and MM* model for 1,2 g = .In 2015, Zhang et al. [12] proposed a new measure for fault diagnosis of the system, namely, g-extra diagnosability, which restrains that every fault-free component has at least ( ) 1 g + fault-free nodes.In [12], they studied the g-extra diagnosability of the n-dimensional hypercube under the PMC model and MM* model.The n-dimensional bubble-sort star graph n BS has many good properties.In 2016, Wang et al. [13]

Preliminaries
In this section, some definitions and notations needed for our discussion, the alternating group graph, the PMC model and the MM* model are introduced.

Notations
A multiprocessor system is modeled as an undirected simple graph ( ) whose vertices (nodes) represent processors and edges (links) represent communication links.Given a nonempty vertex subset V ′ of V, the induced sub- . For neighborhoods and degrees, we will usually omit the subscript for the graph when no confusion arises.A graph G is said to be k-regular if for any vertex v, ( ) of a graph G is the minimum number of vertices whose removal results in a disconnected graph or only one vertex left when G is complete.Let 1 F and 2 F be two distinct subsets of V, and let the symmetric difference ( ) ( ) For graph-theoretical terminology and notation not de- fined here we follow [14].Let ( ) The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G, denoted by ( ) ( ) of g-extra cuts is said to be the g-extra connectivity of G, denoted by ( ) ( ) Proposition 2.1 [15] Let G be a connected graph.Then ( ) ( ) ( ) ( ) [15] Let G be a connected graph.Then ( ) ( ) ( ) ( )

The PMC Model and the MM * Model
Under the PMC model [5] [8], to diagnose a system G, two adjacent nodes in G are capable to perform tests on each other.For two adjacent nodes u and v in ( ) V G , the test performed by u on v is represented by the ordered pair ( ) The outcome of a test ( ) , u v is 1 (resp. 0)if u evaluate v as faulty (resp.fault-free).We assume that the testing result is reliable (resp.unreliable) if the node u is fault-free (resp.faulty).A test assignment T for G is a collection of tests for every adjacent pair of vertices.It can be modeled as a directed testing graph ( ) ( ) , where ( ) ∈ implies that u and v are adjacent in G.The collection of all test results for a test assignment T is called a syndrome.
Formally, a syndrome is a function . The set of all faulty processors in G is called a faulty set.This can be any subset of ( ) V G .For a given syn- drome σ, a subset of vertices ( ) ⊆ is said to be consistent with σ if syndrome σ can be produced from the situation that, for any ( ) , F F is a distinguishable pair.
Using the MM model, the diagnosis is carried out by sending the same testing task to a pair of processors and comparing their responses.We always assume the output of a comparison performed by a faulty processor is unreliable.
The comparison scheme of a system ( ) The MM* model is a special case of the MM model.In the MM* model, all comparisons of G are in the comparison scheme of G, i.e., if ( ) Similar to the PMC model, we can define a subset of vertices ( ) ) For any given system G, ( ) ( ) In a system ( ) if it does not contain all the neighbor vertices of any vertex in G.A system G is conditional t-diagnosable if every two distinct conditional faulty subsets Theorem 2.4 [10] For a system ( ) under the PMC model.Therefore, ( ) ( ) In a system ( ) has more than g nodes.G is g-extra t -diagnosable if and only if for each pair of distinct faulty g-extra vertex subsets ( ) Proposition 2.5 [13] For any given system G, ( ) ( )

Alternating Group Graph
In this section, we give the definition and some properties of the alternating group graph.In the permutation For the conveni- ence, we denote the permutation permutation can be denoted by a product of cycles [18].For example, ( ) . The product στ of two permutations is the composition function τ followed by σ, that is, ( )( ) ( ) . For terminology and notation not defined here we follow [18].
Let [ ] { }  , where every vertex ( ) AG be defined as above.Then there are ( )  Proposition 2.14 [20] For ( ) for some , i j with i j ≠ .

The 2-Extra Diagnosability of Alternating Group Graphs under the PMC Model
In this section, we will give 2-extra diagnosability of alternating group graph networks under the PMC model.  see Figure 1).We prove this lemma (part) by induction on n.The result holds for 4 n = .Assume 5 n ≥ and the result holds for AG has a fixed i in the last position of the label strings which represents the vertices and is isomorphic to Therefore, 1 6 25 6 6 19 F is a 2-extra cut of 1 n AG − , and , by Propositions 2.10 and 2.11, , by Proposition 2.14, , F F is not satisfied with the condition in Theorem 3.1, i.e., there are no edges between ( ) (  ) Without loss of generality, assume that 2 1 \

By the definition of
Since there are no edges between ( ) ( ) is also a 2-extra faulty set when 1 2 \ F F = ∅ .Since there are no edges between ( ) F F ≠ ∅  .Since 5 n ≥ , by Theorem 2.9,  1) There are two vertices ( ) and there is a vertex 2) There are two vertices and there is a vertex ( ) 3) There are two vertices and there is a vertex ( ) Then the 2-extra diagnosability of the n-dimensional alternating group graph n AG under the MM* model, ( ) , and let ( ) , F F is not satisfied with any one condition in Theorem 4.1.Without loss of generality, assume that 2 1 \ . Similar to the discussion on Suppose, on the contrary, that \ u F F ∈ such that u is adjacent to 1 w .Thus, there is just a vertex isolated vertex.Thus, let 1 2 \ F F ≠ ∅ .Similarly, we can deduce that there is just a vertex \ n W A F F ⊆  be the set of isolated vertices in ( ) , and let H be the induced subgraph by the vertex set ( ) ( ) , F F is not satisfied with the condition (1) of Theorem 4.1, and any vertex of ( ) V H is not isolated in H, we induce that there is no edge between ( ) , and let 1 vuw ab be a path in n AG (see Figure 2).
Since there is no edge between ( )

Conclusion
In
of a vertex v is the number of edges incident with v.The minimum degree of a vertex in G is denoted by ( ) of v in G to be the set of vertices adjacent to v. u is called a neighbor vertex or a neighbor of v for is a graph, whose vertex set is V ′ and the edge set is the set of all the edges of G with both endpoints in V ′ .The de-American Journal of Computational Mathematics gree ( )G d v G N v ( ) Furthermore, Proposition 2.12 ([20]) Let F be a vertex-cut of n AG F − satisfies one of the following conditions:1) n AG F − has two components, one of which is an isolated vertex or an American Journal of Computational Mathematics diagnosable under the PMC model if and only if there is an edge uv E 9, ( ) ( ) is not satisfied with any one condition in Theorem 4.1, by the condition (1) of Theorem 4.1, for any pair of this paper, we investigate the problem of 2-extra diagnosability of the n-dimensional alternating group graph n AG under the PMC model and MM* model.It is proved that 2-extra diagnosability of the n-dimensional alternating group graph n AG under the PMC model and MM* model is 6 17 n − , where 5 n ≥ .The above results show that the 2-extra diagnosability is several times larger than the classical diagnosability of n AG depending on the condition: 2-extra.The work will help engineers to develop more different measures of 2-extra diagnosability based on application environment, network topology, network reliability, and statistics related to fault patterns.