A Proof for g-Good-Neighbor Diagnosability of Exchanged Hypercubes

The diagnosability of a multiprocessor system or an interconnection network 
is an important research topic. The system and an interconnection network 
have an underlying topology, which is usually presented by a graph. In this 
paper, we show proof for the g-good-neighbor diagnosability of the exchanged 
hypercube EH (s,t) under the PMC model and MM* model.


Introduction
A multiprocessor system and interconnection network have an underlying topology, which is usually presented by a graph, where nodes represent processors and links represent communication links between processors. Some processors may fail in the system and processor fault identification plays an important role in reliable computing. The identification process is called the diagnosis of the system. 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. [1]. Under the PMC model, 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 [2]. Under the MM model, to diagnose a system, a node sends the same task to two of its neighbors, and then compares their responses. The MM * is a special case of the MM model and each node must test all pairs of its adjacent nodes of the system. The diagnosability of the system is one important study topic. In 2012, Peng et al. [3] proposed measurement for fault diagnosis of the system, namely, the g-good-neighbor diagnosability (which is also called the g-good-neighbor conditional diagnosability), which requires that every fault-free node has at least g fault-free neighbors. Numerous studies have been investigated under the PMC and the MM model or the MM * model, see [2]- [23]. Let

( )
, EH s t be the exchanged hypercube with 1 s t ≤ ≤ . In this paper, we show the following: 1) The g-good-neighbor diagnosability of ( ) , EH s t is ( ) The rest of this paper is organized as follows: In Section 2, we provide the terminology and preliminaries for the system diagnosis. In Section 3, we shall show the g-good-neighbor diagnosability of the exchanged hypercube under the PMC model and the MM * model. Finally, the conclusion is given in Section 4.

Preliminaries
A multiprocessor system and a network are modeled as an undirected simple graph For graph-theoretical terminology and notation not defined here we follow [24].
for every vertex v in \ V F . A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that G F − is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G, denoted by ( ) ( ) A connected graph G is said to be g-good-neighbor connected if G has a g-good-neighbor cut.

The g-Good-Neighbor Diagnosability of the Exchanged Hypercube under the PMC and the MM * Model
Theorem 3.1.
[9] For 1 s t ≤ ≤ and any g with 0 g s . By the proof of Lemma 3.1 in [9], we have the following.

( )
, EH s t be the exchanged hypercube with 1 s t ≤ ≤ . g V is defined as above for 0 g s ≤ ≤ . Then , EH s t be the exchanged hypercube with 1 s t ≤ ≤ and any g with 0 g s ≤ ≤ . Then the g-good-neighbor diagnosability of ( ) Before discussing the g-good-neighbor diagnosability of the exchanged hypercube under the MM * model, we first give two existing results.
and there is a vertex 2) There are two vertices is a minimum g-good-neighbor cut of G. Then the g-good-neighbor diagnosability of G is less than or equal to . Then for any vertex w W ∈ , w are adjacent to 1 v and 2 v .
Note that there are at most two common neighbors for any pair of vertices in ( ) , EH s t , it follows that there are at most two isolated vertices in Suppose that there is exactly one isolated vertex v in

EH s t contains no triangle, it follows that
Thus, Suppose that there are exactly two isolated vertices v and w in ( ) 1 2 , EH s t F F − − . Let 1 v and 2 v be adjacent to v and w, respectively. Then Since the vertex set pair ( )

Conclusion
In this paper, we investigate the problem of the diagnosability of the exchanged hypercube