The Tightly Super 3-Extra Connectivity and Diagnosability of Locally Twisted Cubes

Diagnosability of a multiprocessor system G is one important measure of the reliability of interconnection networks. In 2016, Zhang et al. proposed the g-extra diagnosability of G, which restrains that every component of G S − has at least ( ) 1 g + vertices. The locally twisted cube n LTQ is applied widely. In this paper, we show that n LTQ is tightly ( ) 4 9 n − super 3-extra connected for 6 n ≥ and the 3-extra diagnosability of n LTQ under the PMC model and MM model is 4 6 n − for 5 n ≥ and 7 n ≥ , respectively.

In this paper, we show that n LTQ is tightly ( )

Introduction
At present, semiconductor technology has been widely applied in various fields of large-scale computer systems.But processors or communication links failures of a multiprocessor system give our live a lot of troubles.How to find out the faulty processors accurately and timely becomes the primary problem when the system is in operation.The diagnosis of the system is the process of identifying the faulty processors from the fault-free ones.
There are two well-known diagnosis models, one is the PMC diagnosis model, introduced by Preparata et al. [1] and the other is the MM model, proposed by Maeng and Malek [2].In the PMC model, any two neighbor processors can test each other.In the MM model, to diagnose a system, we can compare their responses after a node sends the same task to its two neighbors.Sengupta and Dahbura [3] suggested a further modification of the MM model, called the MM * model, in which each node must test another two neighbors.
In 2012, Peng et al. [14] proposed a measure for faulty diagnosis of the system, namely, the g-good-neighbor diagnosability, which restrains every fault-free node containing at least g fault-free neighbors.In [14], they studied the g-goodneighbor diagnosability of the n-dimensional hypercube under the PMC model.In 2016, Wang and Han [15] studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the MM * model.In 2016, Zhang et al. [16] proposed the g-extra diagnosability of the system, which restrains that every component of G S − has at least ( ) vertices and showed the g-extra diagnosability of hypercubes under the PMC model and MM * model.Ren et al. [17]

Notations
A multiprocessor system is modeled as an undirected simple graph ( ) whose vertices (nodes) represent processors and edges (links) represent communication links.Suppose that V ′ is a nonempty vertex subset of V.The induced subgraph by V ′ in G, denoted by [ ] G V ′ , is a graph, whose vertex set is V ′ and whose edge set consists of all the edges of G with both endpoints in V ′ .The degree ( ) G d v of a vertex v in G is the number of edges incident with v.We denote by ( ) G δ the minimum degree of vertices of G.For any vertex v, we define the neighborhood ( ) G N v 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 We denote by ( ) . 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 ( ) graph is one whose each edge has one end in subsets of vertex X and one end in subsets of vertex Y; such a partition ( ) , X Y is called a bipartition of the graph.
A complete bipartite graph is a simple bipartite graph with bipartition ( ) F and 2 F be two distinct subsets of V, and let the symmetric difference ( ) ( ) . For graph-theoretical terminology and notation not defined here we follow [20].

Let
( ) The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G, denoted by ( ) ( ) The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G, denoted by ( ) ( ) Proposition 1. ( [21]) Let G be a g-extra and g-good-neighbor connected graph.Then ( ) ( ) ( ) ( ) Proposition 2. ( [21]) Let G be a 1-good-neighbor connected graph.Then

Definitions and Propositions
Definition 3. ( [22] [23] [24] [25]) A system G 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 G is the maximum value of t such that G is t-diagnosable.
For the PMC model and MM * model, we follow [26].Under the PMC model, to diagnose a system , 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 ( ) , u v .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 faultfree(resp.faulty).A test assignment T for G is a collection of tests for every adjacent pair of vertices.The collection of all test results for a test assignment T is called a syndrome.For a given syndrome σ , a subset of vertices ( ) is said to be consistent with σ if syndrome σ can be produced from the situation that, for any ( ) denote the set of all syndromes which F is consistent with.
Under the PMC model, two distinct sets 1 F and 2 Similar to the PMC model, we can define a subset of vertices ( ) V G are indistinguishable (resp.distinguishable) under the MM * model.

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 4. [18] For any given system G, ( ) ( ) For an integer 1 n ≥ , a binary string of length n is denoted by 2 n n − edges, which can be recursively defined as follows [27].Definition 5. ( [27]) For 2 n ≥ , an n-dimensional locally twisted cube, denoted by n LTQ , is defined recursively as follows: 1)
2) For 3 n ≥ , n LTQ is built from two disjoint copies of    2 for all the remaining bits.( )

The Connectivity of Locally Twisted Cubes
and , u v are adjacent, by Propo- sition 7, , u v have no the common neighbor vertex.Similarly, , x w have no the common neighbor vertex and , v w have no the common neighbor vertex.

Since
( ) Since u and x are not adjacent, by proposition 7, ( ) Since , u v are adjacent, by Proposition 7, And since ( ) u w are not adjacent and v is the common neighbor vertex of u and w, by Lemma 3, ( ) ≤ .Since w is the common neighbor vertex of v and x and , v x are not adjacent, by pro- position 7, ( ) Since , u v are adjacent, by Proposition 7, Case 4.
( ) This case is clear.
In conclusion, ( ) ( ) 4 9 Proof.Since ( ) Since , v w are not adjacent and u is a common neighbor vertex of v, w, by Proposition 7, and the definition of n LTQ , we have that ( ) ( ) To prove

5
LTQ F − has three components, two of which are isolated vertices; 2)
( ) . By Lemmas 1 and 2, is connected or has two components, one of which is an isolated vertex.Since ( ) ( ) ( )  ) ( ) satisfies one of the following conditions: 1) has four components, three of which are isolated vertices; 2) has three components, one of which is isolated vertices and one of which is a 2 K ; 3) has three components, two of which are isolated vertices; 4) has two components, one of which is a path of length two; 5) has two components, one of which is an isolated vertex; 6) has two components, one of which is a 2 K ; 7) Thus, F LTQ n − satisfies one of the conditions ( 1)-( 7) in Lemma 10.
Suppose that Suppose that ) Combining this with ( ) ( ) ) . Combining this with 1 3 F ≤ , there is one ( )   ) . We consider the following cases.
( ) By Lemmas 1 and 2, is connected or has two components, one of which is an isolated vertex.Since ( ) satisfies one of the following conditions: 1) has three components, two of which are isolated vertices; 2) has two components, one of which is an isolated vertex; 3) By Lemmas 1 and 2, is connected or has two components, one of which is an isolated vertex.Since 0 3 8 F n = − , by Lemma 10, satisfies one of the following conditions: 1) has four components, three of which are isolated vertices; 2) has three components, one of which is isolated vertices and the other of which is a 2 K ; 3) has three components, two of which are isolated vertices; 4) has two components, one of which is a path of length two; 5) has two components, one of which is an isolated vertex; 6) has two components, one of which is a 2 K ; 7) satisfies the condition (4), i.e., has two components, one of which is a path of length two, denoted by P uvw = , has two components, one of which is an isolated vertex x, and  ) ( ) satisfies one of the following conditions: 1) has four components, three of which are isolated vertices; 2) has three components, one of which is isolated vertices and the other of which is a 2 K ; 3) has three components, two of which are isolated vertices; 4) has two components, one of which is a path of length two; 5) has two components, one of which is an isolated vertex; 6) has two components, one of which is a 2 K ; 7)   ) ( ) If there exists a 3-path P in ) By Corollary 1, ( ) ( ) If there exists a component 1,3 K in If there exists a 4-cycle C in ) has not a 4-cycle.
By Lemma 1, ) ) . Combining this with ( ) ( ) In this case, F is not a minimum 3-extra cut of n LTQ , a contradiction.

The 3-Extra Diagnosability of the Locally Twisted Cube under the PMC Model
In this section, we shall show the 3-extra diagnosability of locally twisted cubes under the PMC model.Proof.Let A be defined in Lemma 7, and let ( ) ( ) Therefore, , F F is not satisfied with the condition in Theorem 10, i.e., there are no edges between ( ) ( ) The following we discuss the case when 2 1 \ Since there are no edges between ( ) ( ) ∩ is also a 3-extra faulty set.Since there are no edges between ( ) F F F ∩ = is also a 3-extra faulty set.Since there are no edges between ( )  1) There are two vertices ( ) and there is a vertex 2) There are two vertices 3) There are two vertices Proof.Let A be defined in Lemma 7, and let ( ) Therefore, Similarly, since 1 2 \ F F ≠ ∅ , by the condition (2) of Theorem 12 and the hypothesis, we can deduce that there is just a vertex  , and H be the induced subgraph by the vertex set ( ) ( ) . Then for any W w ∈ , there are ( ) F F ∩ .By Lemmas 14 and 3, ( ) ( ) Since the vertex set pair ( ) , F F is not satisfied with the condition (1) of Theorem 12, and there are not isolated vertices in H , we induce that there is no edge between ( ) then this is a contradiction to that n LTQ is connected.Therefore,       ) ) graph G is the minimum number of vertices whose removal results in a disconnected graph or only one vertex left.Let 1

1 n
an edge, where "+" represents the modulo 2 addition.The edges whose end vertices in different1 n iLTQ s − are called to be crossedges.Figures 1-3 show four examples of locally twisted cubes.The locally twisted cube can also be equivalently defined in the following non-recursive fashion.Definition 6. ([27]) For 2 n ≥ , the n-dimensional locally twisted cube, denoted by n LTQ , is a graph with { } 0,as the node set.Two nodes

Proposition 7 .
([28]) Let n LTQ be the locally twisted cube.If two vertices , u v are adjacent, there is no common neighbor vertex of these two vertices, i.e., two vertices , u v are not adjacent, there are at most two common neighbor vertices of these two vertices, i.e., components, one is trivial and the other is nontrivial.Lemma 3. ([17]) Let n LTQ be the locally twisted cube.Then all cross-edges of n LTQ is a perfect matching.Lemma 4. ([30]) Let n LTQ be the locally twisted cube.Then − are isomorphic to 1 n LTQ − .Without loss of generality, we have the following cases.

Corollary 1 .
Let n LTQ be the locally twisted cube and let H be a 0 0111, 0 0101, 0 0100 A =     and let n LTQ be the locally twisted cube with 4 n ≥ .If ( ) is connected or has two components, one of which is an isolated vertex.Since 0 6 F = , by Lemma 8, has three components, two of which are isolated vertices; has four components, three of which are isolated vertices; 2) n LTQ F − has three components, one of which is isolated vertices and one of which is a 2 K ; 3) n LTQ F − has three components, two of which are isolated vertices;

(
Then has four components, three of which are isolated vertices; 2) n LTQ F − has three components, one of which is isolated vertices and one of which is a 2 K ; 3) n LTQ F − has three components, two of which are isolated vertices;

(
, one of which is isolated vertices and one of which is a 2 K ; 3) n LTQ F − has three components, two of which are isolated vertices; 4) n LTQ F − has two components, one of which is a path of length two; 5) n LTQ F − has two components, one of which is an isolated vertex;

iV C ≥ . Since 2 F also is a 3 -
extra faulty set of n LTQ , we have that every component i C′ of

2 1 \FF
≠ ∅ and ( ) V H ≠ ∅ .The proof of Claim 1 is complete.not satisfied with any one condition in Theorem 12, by the condition (1) of Theorem 12, for any pair of adjacent vertices

(
studied the tightly super 2-extra connectivity and 2-extra diagnosability of locally twisted cubes n LTQ .In 2016, Wang et al. [18] studied the 2-extra diagnosability of the bubble-sort star graph n BS under the PMC model and MM * model.In 2017, Wang and Yang [19] studied the 2-good-neighbor (2extra) diagnosability of alternating group graph networks under the PMC model and MM * model. Thus, we have that

The 3-Extra Diagnosability of the Locally Twisted Cube under the MM * Model
model is more than or equal to 4 6 n − , i.e., 7 n ≥ .Then the 3-extra diagnosability of the locally twisted cube n LTQ under the MM * Proof.By the definition of the 3-extra diagnosability, it is sufficient to show that n LTQ is 3-extra ( ) 4 6 n − -diagnosable.By Theorem 12, suppose, by way of contradiction, that there are two distinct