Fault diagnosability of data center networks

0.23 MB·

by  Mei-Mei Gu

Fault diagnosability of data center networks
Mei-Mei Gua, Rong-Xia Haoa,∗, Shuming Zhoub
aDepartment of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
bSchool of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350108, China
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Abstract
n
a
The data center networks D , proposed in 2008, has many desirable features such as
J n,k
high network capacity. A kind of generalization of diagnosability for network G is g-good-
9
2 neighbor diagnosability which is denoted by tg(G). Let κg(G) be the Rg-connectivity. Lin
et. al. in [IEEE Trans. on Reliability, 65 (3) (2016) 1248–1262] and Xu et. al in [Theor.
]
C Comput. Sci. 659(2017)53–63]gavethesameproblemindependentlythat: therelationship
D between the Rg-connectivity κg(G) and t (G) of a general graph G need to be studied in
g
. the future. In this paper, this open problem is solved for general regular graphs. We firstly
s
c establish the relationship of κg(G) and t (G), and obtain that t (G) = κg(G) +g under
g g
[
some conditions. Secondly, we obtain the g-good-neighbor diagnosability of Dk,n which are

1 t (D ) = (g +1)(k −1)+n+g for 1 ≤ g ≤ n−1 under the PMC model and the MM
g k,n
v
model, respectively. Further more, we show that D is tightly super (n+k−1)-connected
9 k,n
5 for n ≥ 2 and k ≥ 2 and we also prove that the largest connected component of the survival
2 graphcontains almostalloftheremainingvertices inD when2k+n−2vertices removed.
k,n
0
0 Keywords: Data center network; g-good-neighbor diagnosability; PMC model; MM
.
2 model; Fault-tolerance.
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: 1. Introduction
v
i
X The study of interconnection networks has been an important research area for parallel
r and distributed computer systems. A network can be modeled as a graph, in which vertices
a
and edges correspond to processors and communication links, respectively. Network relia-
bility is one of the major factors in designing the topology of an interconnection network.
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 as system-level diagnosis. For the purpose of self-diagnosis of a system, a
numberofmodelshavebeenproposedfordiagnosingfaultyprocessorsinanetwork. Among
the proposed models, PMC model [26] and comparison model (MM model) [24] are widely
used. In the PMC model, every processor can test the processor that is adjacent to it and
only the fault-free processor can guarantee reliable outcome. In the MM model, to diagnose
the system, a processor sends the same task to one pair of its neighbors, and then compares
their responses. 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
∗Corresponding author
Email addresses: [email protected], (Mei-Mei Gu),[email protected], (Rong-XiaHao),
[email protected], (ShumingZhou)
Preprint submitted to Elsevier February 2, 2017
maximum number of faulty processors which can be correctly identified. In 2005, Lai et
al. [19] introduced a restricted diagnosability of the system called conditional diagnosability
by assuming that it is impossible that all neighbors of one vertex are faulty simultaneously.
The diagnosabilities and conditional diagnosabilities of many networks are studied in liter-
atures [1]-[3],[11]-[14], [15], [17]-[18], [21], [22], [28], [37]etc. Inspiredby this concept, Peng
et al. [25] then proposed the g-good-neighbor diagnosability, which requires every fault-free
vertex has at least g fault-free neighbors.
Definition 1. AfaultsetF ⊆ V(G)isag-good-neighbor faulty setif|N (v)∩(V(G)\F)| ≥
G
g for every vertex v ∈ V(G)\F. A g-good-neighbor cut of a graph G is a g-good-neighbor
faulty set F such that G−F is disconnected. For an arbitrary graph G, g-good-neighbor
cuts do not always exist for some g. A graph G is called an Rg-graph if it contains at least
one g-good-neighbor cut. For an Rg-graph G, the minimum cardinality of g- good-neighbor
cuts is said to be the Rg-connectivity of G, denoted by κg(G). The parameter κ1(G) is
equal to extra connectivity κ (G) which is proposed by F´abrega and Fiol [10], where κ (G)
1 k
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.
Definition 2. A system G = (V,E) is g-good-neighbor t-diagnosable if F and F are
1 2
distinguishable (the definition of distinguishable is in Section 2), for each distinct pair of
g-good-neighbor faulty sets F and F of V with |F | ≤ t and |F |≤ t. The g-good-neighbor
1 2 1 2
diagnosability t (G) of a graph G is the maximum value of t such that G is g-good-neighbor
g
t-diagnosable.
Theclassical diagnosability reliesonanassumptionthatallneighborsofeach vertex ina
parallelsystemcanpotentiallyfailatthesametime. Buttheg-good-neighbordiagnosability
is superior to the classical diagnosability in terms of measuring diagnosability for large-
scale parallel systems. The problem of determining the g-good-neighbor diagnosability for
g = 1,2 of numerous networks, for examples, see [29] and [30], has received much attention
in recent years. But little is known about t (G) with a general non-negative integer g for
g
networks except for hypercubes, k-ary n-cubes etc. Peng et al. [25] showed that the g-
good-neighbor diagnosability of the n-dimensional hypercube Q under the PMC model is
n
2g(n−g)+2g −1 for 0≤ g ≤ n−3. Yuan et al. [35] and [36] studied the g-good-neighbor
diagnosability of the k-ary n-cubes (k ≥ 4) and 3-ary n-cubes, respectively, underthe PMC
model and MM model. Wang and Han [32] determined the g-good-neighbor diagnosability
of the n-dimensional hypercube Q under the MM model.
n
Xu et al. [23] and Lin et al. [34] gave the same problem independently that the rela-
tionship between the Rg-connectivity κg(G) and t (G) of a general graph G need
g
to be studied in the future.
In this paper, we firstly study the relation between g-good-neighbor diagnosability and
Rg-connectivity for regular graphs and obtain the following Theorem 1. Secondly, we prove
that D is tightly super (n+k−1)-connected for n ≥ 2 and k ≥ 2 and we also prove that
k,n
the largest connected component of the survival graph contains almost all of the remaining
vertices in D when almost 2k + n − 2 vertices are removed. Thirdly, we obtain that
k,n
the g-good-neighbor diagnosability of D which are t (D ) = (g + 1)(k − 1) + n + g
k,n g k,n
for 1 ≤ g ≤ n − 1 under the PMC model and the MM model, respectively. As direct
corollaries,theg-good-neighbordiagnosabilityofthe(n,k)-starnetworksS andthe(n,k)-
n,k
arrangement graphs A are obtained.
n,k
2
Theorem 1. Let n, g and N be non-negative integers. Let G be an n-regular connected Rg-
graph with order N. Suppose G has a complete subgraph K of order m, where m ≤ n−1.
m
Let κg(G) be the Rg-connectivity of G. If G satisfies the conditions (1) and (2) under the
PMC model; or G satisfies the conditions (1),(2) and (3) under the MM model.
(1) there exists a minimum g-good-neighbor cut T such that G−T has exactly two com-
ponents, one of which is isomorphic to K , where g ≤ m−1;
g+1
(2) N ≥ 2κg(G)+3κ1(G)+2g−n−1;
(3) for any F ⊆ V(G) and |F| ≤ κ1(G), G−F is eitherconnected; or has two components,
one of which is a trivial component; or has two components, one of which is an edge;
or has three components, two of which are trivial components.
Then, t (G) = κg(G)+g for 1 ≤ g ≤ m−1.
g
Theremainderofthispaperisorganized as follows. Section 2introducessomenecessary
notations and basic lemmas. Our main results are given in Section 3. As applications of
ourmain result,Section 4concentrates on theg-good-neighbor diagnosability of threekinds
of graphs: data center networks D , the (n,k)-star networks S , the (n,k)-arrangement
n,k n,k
graphs A . Section 5 concludes the paper.
n,k
2. Preliminaries
In this section, we give some terminologies and notations of combinatorial network
theory. We follow [33] for terminologies and notations not defined here.
Weuseagraph,denotedbyG = (V(G),E(G)), torepresentaninterconnection network,
where a vertex u∈ V(G) represents a processor and an edge (u,v) ∈ E(G) represents a link
between vertices u and v. Two vertices u and v are adjacent if (u,v) ∈ E(G), the vertex u
is called a neighbor of v, and vice versa. For a vertex u ∈ V(G), let N (u) denote a set of
G
vertices inGadjacenttou. Thecardinality |N (u)|representsthedegree ofuinG, denoted
G
by d (u) (or simply d(u)), δ(G) the minimum degree of G. For a vertex set U ⊆ V(G),
G
the neighborhood of U in G is defined as N (U) = N (v)−U. If |N (u)| = k for any
G G G
vS∈U
vertex in G, then G is k-regular. Let G be a connected graph, if G−S is still connected
for any S ⊆ V(G) with |S| ≤ k−1, then G is k-connected. A subset S ⊆ V(G) is a vertex
cut if G−S is disconnected. The connectivity of a graph G, denoted by κ(G), defined as
the minimum number of vertices whose removal results in a disconnected or trivial graph.
A k-regular graph is loosely super k-connected if any one of its minimum vertex cuts is a
set of the neighbors of some vertex. If, in addition, the deletion of a minimum vertex cut
results in a graph with two components (one of which has only one vertex), then the graph
is tightly super k-connected. A graph H is a subgraph of a graph G if V(H) ⊆ V(G) and
E(H) ⊆ E(G). The components of a graph G are its maximally connected subgraphs. A
component is trivial if it has only one vertex; otherwise, it is nontrivial.
To diagnose faults, a number of tests are performed on vertices. The collection of all
test results is called a syndrome. Let F be a subset of V(G). F is said to be compatible
with a syndrome σ if σ can arise from the circumstance that all vertices in F are faulty
and all vertices in V(G)\F are fault free. A system is said to be diagnosable if, for every
syndromeσ, there is a uniqueF ⊆ V(G) such that F is compatible with σ. Let σ = {σ : σ
F
is compatible with F}. Two distinct subsets F ,F ⊆ V(G) are said to be indistinguishable
1 2
3
if and only if σ ∩σ 6=∅; otherwise, F ,F are said to be distinguishable. The symmetric
F1 F2 1 2
difference of F ⊆ V(G) and F ⊆ V(G) is defined as the set F ∆F = (F \F )∪(F \F ).
1 2 1 2 1 2 2 1
The following two lemmas characterize a graph for g-good-neighbor t-diagnosable under
the PMC model and the MM model, respectively.
Lemma 1. ([27, 35]) A system G = (V,E) is g-good-neighbor t-diagnosable under the PMC
model if and only if there is an edge (u,v) ∈ E with u ∈ V \(F ∪F ) and v ∈ F ∆F for
1 2 1 2
each distinct pair of g-good-neighbor faulty sets F and F of V with |F |≤ t and |F | ≤ t.
1 2 1 2
Lemma 2. ([8, 35]) A system G = (V,E) is g-good-neighbor t-diagnosable under the MM
model if and only if for each distinct pair of g-good-neighbor faulty sets F and F of V with
1 2
|F | ≤ t and |F |≤ t satisfies one of the following conditions.
1 2
(1) There are two vertices u,w ∈ V \(F ∪F ) and there is a vertex v ∈ F ∆F such that
1 2 1 2
(u,v) ∈ E and (u,w) ∈ E.
(2) There are two vertices u,v ∈ F \F and there is a vertex w ∈ V \(F ∪F ) such that
1 2 1 2
(u,w) ∈ E and (v,w) ∈ E.
(3) There are two vertices u,v ∈ F \F and there is a vertex w ∈ V \(F ∪F ) such that
2 1 1 2
(u,w) ∈ E and (v,w) ∈ E.
3. Proof of Theorem 1
Proof. First, we prove t (G) ≤ κg(G)+g under the PMC and the MM model.
g
Let T be the minimum g-good-neighbor cut of G satisfies Condition (1), i.e. G−T has
two components, one of which is isomorphic to K , say A. Clearly, T = N (A), and
g+1 G
δ(G − T) ≥ g. Let F = N (A), F = N (A) ∪ A, see Figure 1. Then |F | = κg(G),
1 G 2 G 1
|F | = κg(G) + g + 1, δ(G − F ) ≥ g and δ(G − F ) ≥ g. It implies F and F are g-
2 1 2 1 2
good-neighbor faulty sets of G. Note that F ∆F = A, N (A) = F ⊆ F , there is no
1 2 G 1 2
edge of G between V(G)\(F ∪F ) and F ∆F . By Lemma 1, G is not g-good-neighbor
1 2 1 2
(κg(G)+g+1)-diagnosable under the PMC model, so t (G) ≤ κg(G)+g under the PMC
g
model.
Note that F \ F = ∅, F \ F = A, F and F do not satisfy any one condition in
1 2 2 1 1 2
Lemma 2. By Lemma 2, G is not g-good-neighbor (κg(G)+g +1)-diagnosable under the
MM model, so t (G) ≤ κg(G)+g under the MM model.
g
Figure 1: The illustration of Theorem 1
Next we prove t (G) ≥ κg(G)+g, i.e., G is g-good-neighbor (κg(G)+g)-diagnosable.
g
(I) For the PMC model, it is equivalent to prove Claim 1.
4
Claim 1. For each distinct pair of g-good-neighbor faulty sets F and F of G with |F | ≤
1 2 1
κg(G)+g and |F | ≤ κg(G)+g, there is an edge (x,y) ∈ E(G) with x ∈ V(G)\(F ∪F )
2 1 2
and y ∈F ∆F .
1 2
Proof of Claim 1. Suppose, on the contrary, that there are two distinct g-good-neighbor
faulty sets F and F of G with |F | ≤ κg(G)+g and |F | ≤ κg(G)+g, there is no edge
1 2 1 2
between V(G)\(F ∪F ) and F ∆F .
1 2 1 2
Without loss of generality, assume that F \ F 6= ∅. If V(G) = F ∪ F , then N =
2 1 1 2
|V(G)| = |F ∪ F | = |F | + |F | − |F ∩ F | ≤ 2κg(G) + 2g < N, it is a contradiction.
1 2 1 2 1 2
Therefore, V(G) 6= F ∪F .
1 2
Note that F is a g-good-neighbor faulty set, δ(G−F ) ≥ g. Because there exists no
1 1
edge between V(G)\(F ∪F ) and F ∆F , δ(G−(F ∪F )) ≥ g and δ(G[F \F ]) ≥ g.
1 2 1 2 1 2 2 1
Similarly, δ(G[F \F ]) ≥ g if F \F 6= ∅. Thus, F ∩F is a g-good-neighbor cut because
1 2 1 2 1 2
of F \F 6= ∅ and G−(F ∪F ) 6= ∅, so |F ∩F | ≥ κg(G). Note that δ(G[F \F ]) ≥ g,
2 1 1 2 1 2 2 1
it follows that |F \F | ≥ g+1. Then, |F | = |F \F |+|F ∩F | ≥ κg(G)+g+1, which
2 1 2 2 1 1 2
contradicts with |F | ≤ κg(G)+g. The proof of Claim 1 is completed.
2
(II) Now we consider the MM model. We prove t (G) ≥ κg(G)+g, i.e., G is g-good-
g
neighbor (κg(G)+g)-diagnosable.
Suppose, on the contrary, that there are two distinct g-good- neighbor faulty sets F
1
and F of G with |F | ≤ κg(G) + g and |F | ≤ κg(G) + g, but (F ,F ) does not satisfy
2 1 2 1 2
any one condition in Lemma 2. Clearly, |F ∩F | ≤ κg(G) + g − 1 because of F 6= F .
1 2 1 2
Without loss of generality, assume that F \F 6= ∅. If V(G) = F ∪F , then N = |V(G)| =
2 1 1 2
|F ∪F |= |F |+|F |−|F ∩F |≤ 2κg(G)+2g, itisimpossiblebyCondition(2). Therefore,
1 2 1 2 1 2
V(G) 6= F ∪F .
1 2
Claim 2. G−(F ∪F ) has no trivial component.
1 2
Proof of Claim 2. If g = 1, it implies that |F | ≤ κ1(G)+1, |F | ≤ κ1(G)+1 and |F ∩F |≤
1 2 1 2
κ1(G). Let W be the set of trivial components in G−(F ∪F ) and C = G−(F ∪F ∪W).
1 2 1 2
Assume |W| 6= 0. Then F \F 6= ∅ and F \F 6= ∅. For any w ∈ W, note that F (resp.
1 2 2 1 1
F ) is a 1-good-neighbor faulty set, by Lemma 2, there is exactly one vertex u ∈ F \F
2 2 1
(resp. v ∈ F \F ) such that u (resp. v) is adjacent to w.
1 2
Note that F \F 6= ∅, then w has n−2 neighbors in F ∩F , it implies that |F ∩F |≥
1 2 1 2 1 2
n−2. One has |NG[F1∩F2](w)| = |W|(n−2) ≤ dG(v) = n|F1∩F2| ≤ nκ1(G), so
wP∈W v∈FP1∩F2
|W|≤ nκ1(G) ≤ 3κ1(G). If C = ∅, then |V(G)| = |F ∪F |+|W|= |F |+|F |−|F ∩F |+
n−2 1 2 1 2 1 2
|W|≤ 2κg(G)+3κ1(G)+2g < N which contradicts with Condition (2). Thus,C 6= ∅. Note
that (F ,F ) does not satisfy the Condition (1) in Lemma 2 and C is the set of non-trivial
1 2
components of G−(F ∪F ), so there is no edge between C and F ∆F . It implies that
1 2 1 2
F ∩F is a vertex-cut of G and δ(G−(F ∩F )) ≥ 1, i.e., F ∩F is a 1-good-neighbor cut
1 2 1 2 1 2
of G, so |F ∩F | ≥ κ1(G). Since |F ∩F |≤ κ1(G), it implies |F ∩F | = κ1(G).
1 2 1 2 1 2
Note that neither F \ F nor F \ F is empty, so |F \ F | = |F \ F | = 1. Let
1 2 2 1 2 1 1 2
F \F = {v }, F \F = {v }. For any w ∈ W, w is adjacent to both v and v .
1 2 1 2 1 2 1 2
Note that |F ∩F | = κ1(G) andF ∩F is a1-good-neighbor cutof G, by Condition (3),
1 2 1 2
G−(F ∩F ) has two components, one of which is an edge. It follows that v is adjacent
1 2 1
to v and |W|= 0, which contradicts with W 6= ∅.
2
Now we assume that 2 ≤ g ≤ k −1. Since F is a g-good-neighbor faulty set, for any
1
x ∈ G−F1, |NG−F1(x)| ≥ g. As the vertex set pair (F1,F2) is not satisfied with any one
5
condition in Lemma 2. By Condition (3) in Lemma 2, any vertex w ∈ V(G)\(F ∪F )
1 2
has at most one neighbor in F \ F , it implies that |N (w)| ≥ g − 1 ≥ 1, i.e.,
2 1 G−(F1∪F2)
G−(F ∪F ) has no trivial component. The Claim is completed.
1 2
Let y ∈ V(G)\(F ∪F ). By Claim 2, y has at least one neighbor in G−(F ∪F ).
1 2 1 2
Note that the vertex set pair (F ,F ) does not satisfy any one condition in Lemma 2, y has
1 2
no neighbor in F ∆F . By the arbitrary of y, there is no edge between V(G)\(F ∪F )
1 2 1 2
and F ∆F .
1 2
Since F \F 6= ∅, and F is a g-good-neighbor faulty set and condition (3) of Lemma 2,
2 1 1
δ(G[F \F ]) ≥ g. Similarly, δ(G[F \F ]) ≥ g if F \F 6= ∅. Since V(G)−(F ∪F ) 6= ∅
2 1 1 2 1 2 1 2
and F \ F 6= ∅, F ∩ F is a g-good-neighbor cut of G, so |F ∩ F | ≥ κg(G). Since
2 1 1 2 1 2
δ(G[F \F ]) ≥ g, it follows that |F \F | ≥ g +1. Then, |F | = |F \F |+|F ∩F | ≥
2 1 2 1 2 2 1 1 2
κg(G)+g +1, which contradicts with |F | ≤ κg(G)+g. Therefore, G is g-good-neighbor
2
(κg(G)+g)-diagnosable under the MM model and t (G) ≥κg(G)+g.
g
By the above discussion, t (G) = κg(G)+g. The proof is completed.
g
4. Applications
4.1. Application to data center network D
k,n
Guo et al. [12] proposed a server-centric data center network called DCell. Data center
networksD havebeenbecomingmoreandmoreimportantwiththedevelopmentofcloud
k,n
computing.
Given a positive integer m, we use hmi and [m] to denote the sets {0,1,2,...,m}
and {1,2,...,m}, respectively. For any integers k ≥ 0 and n ≥ 2, we use D denote a k-
k,n
dimensionalDCellwithn-portswitches. D is acomplete graphonnvertices. Weuset
0,n k,n
to denote the number of vertices in Dk,n with t0,n = n and ti,n = ti−1,n×(ti−1,n+1), where
i ∈[k]. LetI0,n = hn−1iandIi,n = hti−1,niforanyi ∈ [k]. Then,letVk,n = {ukuk−1···u0 :
ui ∈ Ii,n and i ∈ hki}, and Vkℓ,n = {ukuk−1···uℓ : ui ∈ Ii,n and i ∈ {ℓ,ℓ +1,...,k} for
any ℓ ∈ [k]}. Clearly, |Vk,n| = tk,n and |Vkℓ,n| = tk,n/tℓ−1,n. The definition of Dk,n is as
follows [12].
Definition 3. Dk,n is a graph with vertex set Vk,n, where a vertex u= ukuk−1···ui···u0
is adjacent to a vertex v = vkvk−1···vi···v0 if and only if there is an integer ℓ with
(1) ukuk−1···uℓ = vkvk−1···vℓ,
(2) uℓ−1 6= vℓ−1,
ℓ−2 ℓ−2
(3) uℓ−1 = v0+ (vj ×tj−1,n) and vℓ−1 = u0+ (uj ×tj−1,n)+1 with ℓ > 1;
jP=1 jP=1
k−1 k−1
Or uk 6= vk, uk ≤ vk and uk = v0+ (vj ×tj−1,n) and vk = u0+ (uj ×tj−1,n)+1.
jP=1 jP=1
D is an edge; D is a cycle of length 6. D is shown in Figure 2. It is clear that
0,2 1,2 2,2
D is a regular graph with t vertices.
k,n k,n
When all three conditions of Definition 3 hold, we define that two adjacent vertices u
and v have a leftmost distinct element at position ℓ−1. For any integer d ≥ 0, when two
adjacent vertices u and v have a leftmost differing element at the position d, denoted by
ldiff(u,v) = d. For any α ∈ Vℓ withℓ ∈ [k], weuseDα to denotethegraphobtained by
k,n ℓ−1,n
6
Figure 2: The illustration of D2,2
prefixing the label of each vertex of one copy of Dℓ−1,n with α. Clearly, Dℓ−1,n ∼= Dℓα−1,n.
For any integers n ≥ 2 and k ≥ 1, edges joining vertices in the same copy of Dk−1,n
are called internal edges and edges joining vertices in disjoint copies of Dk−1,n are called
external edges. Clearly, each vertex of Di is joined to exactly one external edge and
k−1,n
(n+k−2)-internal edges for each i ∈ I .
k,n
From the definition of D in [12], the following properties 1 can be gotten directly.
k,n
Proposition 1. Let D be the data center network with k ≥ 0 and n≥ 2.
k,n
(1) D is a complete graph with n vertices labeled as 0,1,2,...,n−1 respectively.
0,n
(2) For k ≥ 1, Dk,n consists of tk−1,n +1 copies of Dk−1,n, denoted by Dki−1,n, for each
u v
i ∈ htk−1,ni. ForanytwocopiesDk−k1,nandDkk−1,n ofDk−1,n withuk ≤ vk,thereexists
u
only one edge (u,v), where u = ukuk−1uk−2···u0 in Dk−k1,n and v = vkvk−1··· ,v0 in
k−1 k−1
v
Dkk−1,n which satisfy that uk = v0+ (vj×tj−1,n) and vk = u0+ (uj×tj−1,n)+1.
jP=1 jP=1
u u
It implies that each vertex in D k has only one neighbor which is not in D k ,
k−1,n k−1,n
called extra neighbor.
(3) For any two distinct vertices u,v in Di , N (u) ∩ N (v) = ∅ and
k−1,n DIk,n\{i} DIk,n\{i}
k−1,n k−1,n
|N (u)| = 1. There is only one edge between Di and Dj for any
DIk,n\{i} k−1,n k−1,n
k−1,n
i,j ∈ I and i6= j.
k,n
Lemma 3. ([12]) The connectivity of D is κ(D ) = n+k−1. For any integers k ≥ 0
k,n k,n
and n≥ 2, the number of vertices in D satisfies t ≥ (n+ 1)2k − 1.
k,n k,n 2 2
Lemma 4. ([31]) For any integers k ≥ 1, n ≥ 2, and n−1 ≥ g, if each fault-free vertex
has at least g fault-free neighbor(s) in D , then there exists a complete graph A of order
k,n
g +1 in D such that N (A) = (g +1)(k −1)+n, and D −N (A) has exactly
k,n Dk,n k,n Dk,n
two components: one is A and the other is D − N (A) − A, where every vertex of
k,n Dk,n
D −N (A)−A has at least g fault-free neighbor(s) in D −N (A)−A.
k,n Dk,n k,n Dk,n
7
Lemma 5. ([31]) For any integer n ≥ 2,
(g+1)(k−1)+n if 0 ≤g ≤ n−1 and k ≥ 1;
κg(D ) =
k,n
(cid:26) (n+k−g−1)th−n+1,n if n ≤ g ≤ n+k−2 and k ≥ 2
Let F be the subset of V(D ). Let F = F ∩Di , f = |F | for i ∈ I , I = {i ∈
k,n i k−1,n i i k,n
I : f ≥ n+k −2}, F = F , J = I \I, F = F and DJ = G[ Dj ]
k,n i I i k,n J j k−1,n k−1,n
i∈I j∈J j∈J
S S S
j
which is the induced subgraph by V(D ). The following Claim 3 is useful.
k−1,n
j∈J
S
Claim 3. ([31]) Let F be a faulty vertex set of D . If |F| ≤(g+1)(k−1)+n with k ≥ 2,
k,n
n ≥ 2 and 0 ≤ g ≤ n−1, then |I| ≤ g+1 and DJ −F is connected.
k−1,n J
Lemma 6. D is tightly super (n+k−1)-connected for n ≥ 2 and k ≥ 2.
k,n
Proof. Note that κ(D )= n+k−1, let F bethe subset of V(D ) with |F| = n+k−1
k,n k,n
and D −F is disconnected. Recall that F = F ∩Di , f = |F | for i ∈ I , I = {i ∈
k,n i k−1,n i i k,n
I : f ≥ n+k−2}, F = F , J = I \I, F = F and DJ = G[ Dj ].
k,n i I i k,n J j k−1,n k−1,n
iS∈I jS∈J jS∈J
By Claim 3, |I| ≤ 1 and DJ −F is connected. We consider the following two cases.
k−1,n J
Case 1. |I| = 0.
In this case, J = I , D −F = DJ −F is connected, which leads to a contradic-
k,n k,n k−1,n J
tion.
Case 2. |I| = 1.
Without loss of generality, let I = {1}, so J = I \{1}, DJ −F is connected. If
k,n k−1,n J
Dk1−1,n−F1 isconnected, since|V(Dk1−1,n)|= tk−1,n ≥ (n+12)2k−1−21 > n+2k−2= |F|for
n ≥ 2 and k ≥ 2, it implies at least one vertex of D1 −F is connected to DJ −F .
k−1,n 1 k−1,n J
As a result, D −F is connected, which leads to a contradiction. In the following, assume
k,n
D1 −F is disconnected.
k−1,n 1
Subcase 2.1. f = n+k−2.
1
LetubetheuniquevertexinF\F . BythesimilardiscussionasCase1, D −D1 −
1 k,n k−1,n
{u} is connected. By Proposition 1, any non-trivial component of D1 −F is connected
k−1,n 1
to D −D1 −{u}. There is exactly one trivial component because |F \F | = 1. Thus,
k,n k−1,n 1
ifD −F isdisconnected, ithasexactly two components,oneofwhichhasonlyonevertex,
k,n
say v, and its only disconnecting set is the set of the neighbors of v.
Subcase 2.2. f = n+k−1.
1
Consequently, F = ∅. Note that each vertex in D1 − F is adjacent to exactly
J k−1,n 1
one vertex in DJ − F = DJ , it implies D − F is connected, which leads to a
k−1,n J k−1,n k,n
contradiction.
Hence, D is tightly (n+k−1)-super connected for n ≥ 2 and k ≥ 2.
k,n
Lemma 7. Let F ⊆ V(D ) and |F| ≤ n with n ≥ 2. Then D −F either is connected;
1,n 1,n
or has two components, the smaller one, say C, C ∈ {K : 1 ≤ t ≤ n}, where K is the
t t
complete graph with order t.
Proof. If n = 2, note that D is a cycle of length 6, it is not different to check the
1,2
result holds. We consider n ≥ 3 as follows. Assume that D − F is disconnected and
1,n
C ,C ,...,C are the disjoint connected components of D −F.
1 2 m 1,n
8
j
For i ∈ [m], C is contained in some subgraph, say D for j ∈ I . If this is not
i 0,n 1,n
true, let T = {x ∈ I : C ∩Dx 6= ∅} and |T| ≥ 2. Note that Dx is a complete graph,
1,n i 0,n 0,n
|V(Cx)| = |Dx |−f . Asthereisexactly onecrossedgebetween Dx andDy , toseparate
i 0,n x 0,n 0,n
C fromother part,ithasatleast f +|V(C )|−|T|= (n−|V(Cx)|)+|V(C )|−|T| =
i x i i i
x∈T x∈T
P P
|T|(n−1) ≥ 2n−2 > n for n ≥ 3 faulty vertices, which is a contradiction.
If C ∼= K ∈ Dj , to separate C from Dj , it has to remove n−t vertices. As every
i t 0,n i 0,n
j
vertex of C has exactly one cross edge connecting to D − D , it need to remove t
i 1,n 0,n
vertices in N (C ), it implies there are no surplus faulty vertices in F. This means
D1,n−D0j,n i
thatm = 1, andC ∼= K istheonlyconnected componentexceptforthelargestcomponent
1 t
in D −F.
1,n
By Lemma 7, D is not tightly super n-connected for n≥ 2.
1,n
Lemma 8. Let F ⊆ V(D ) and |F| ≤ n+2 with n ≥ 2. Then D −F either isconnected;
2,n 2,n
or has two components, one of which is a trivial component; or has two components, one
of which is an edge; or has three components, two of which are trivial components.
Proof. Recall that I = {i ∈ I : f ≥ n}, J = I \ I, and DJ = G[ Dj ], by
2,n i 2,n 1,n 1,n
j∈J
S
Claim 3, |I| ≤2 and DJ −F is connected. We consider the following three cases.
1,n J
Case 1. |I| = 0.
In this case, J = I , D −F = DJ −F is connected.
2,n 2,n 1,n J
Case 2. |I| = 1.
Without loss of generality, let I = {1}, so J = I \{1}, DJ −F is connected. If
2,n 1,n J
D1 −F is connected, since |V(D1 )| = t = n(n+1)> n+2≥ |F| for n ≥ 2, it implies
1,n 1 1,n 1,n
at least one vertex of D1 −F is connected to DJ −F , D −F is connected. In the
1,n 1 1,n J 2,n
following, assume D1 −F is disconnected.
1,n 1
Note that f = |F|−f ≤ 2, by Proposition 1, at most two vertices in D1 −F are
J 1 1,n 1
disconnected with DJ −F . Hence, if D −F is disconnected, then it contains a large
k−1,n J 2,n
component and smaller components which contain at most two vertices in total.
Case 3. |I| = 2.
Without loss of generality, let I = {0,1}, f ≥ n and f ≥ n. Since n + 2 ≥ |F| ≥
0 1
f +f ≥ 2n, i.e. n ≤ 2, so n = 2, f = 2, f = 2 and f = 0.
0 1 0 1 J
Note that f = 0, any component of Di −F with more than one vertex is adjacent
J 1,2 i
to DJ = DJ −F , by Proposition 1, at most one trivial component of Di −F can be
1,n 1,n J 1,2 i
disconnected with DJ −F . It leads to if D −F is disconnected, then it contains a large
1,n J 2,n
component and a trivial component.
Lemma 9. Let F ⊆ V(D ) and |F| ≤ 2k+n−2 with k ≥ 2 and n ≥ 2. Then D −F
k,n k,n
either is connected; or has two components, one of which is a trivial component; or has
two components, one of which is an edge; or has three components, two of which are trivial
components.
Proof. We prove the lemma by the induction on k. By Lemma 8, the result holds for
k = 2. Assume k ≥ 3 and the result holds for Dk−1,n. We consider Dk,n as follows. Recall
that I = {i ∈ I : f ≥ n+k−2}, J = I \I, and DJ = G[ Dj ], by Claim 3,
k,n i k,n k−1,n k−1,n
j∈J
S
|I| ≤ 2. We need only consider the following three cases with respect to I.
9
Case 1. |I| = 0.
In this case, J = I , D −F = DJ −F is connected.
k,n k,n k−1,n J
Case 2. |I| = 1.
Without loss of generality, let I = {1}, so J = I \{1}, DJ −F is connected. If
k,n k−1,n J
Dk1−1,n−F1 is connected, since |V(Dk1−1,n)| = tk−1,n ≥ (n+ 12)2k−1 > n+2k−2 ≥ |F| for
n ≥ 2 and k ≥ 3, it implies at least one vertex of D1 −F is connected to DJ −F .
k−1,n 1 k−1,n J
As a result, D −F is connected. In the following, assume D1 −F is disconnected.
k,n k−1,n 1
Subcase 2.1. n+k−2 ≤ f ≤ 2k+n−4.
1
By inductive hypothesis in D1 , if D1 − F is disconnected, then it contains a
k−1,n k−1,n 1
large component, say B, and smaller components which contain at most two vertices in
total. Since |V(Dk1−1,n)|−2 = tk−1,n −2 ≥ (n + 12)2k−1 − 21 −2 > n+2k −2 ≥ |F| for
n ≥ 2 and k ≥ 3, it implies that B is connected to DJ −F . Note that if D −F
k−1,n J k,n
is disconnected, then D −F contains a large component and smaller components which
k,n
contain at most two vertices in total.
Subcase 2.2. f = 2k+n−3.
1
In this case, |F | = |F|−f ≤ 1. Note that each vertex in D1 is adjacent to exactly
J 1 k−1,n
one vertex in DJ , at most one vertex are disconnected with DJ − F . Thus, if
k−1,n k−1,n J
D −F is disconnected, then it has two components, one of which is a trivial component.
k,n
Subcase 2.3. f = 2k+n−2.
1
Consequently, F = ∅. Note that each vertex in D1 −F is adjacent to exactly one
J k−1,n 1
vertex in DJ −F = DJ , it leads to D −F is connected.
k−1,n J k−1,n k,n
Case 3. |I| = 2.
Without loss of generality, let I = {1,2} and f ≥ f ≥ n+k−2, so J = I \{1,2},
1 2 k,n
DJ −F is connected.
k−1,n J
We Claim f = n+k −2 for i ∈ {1,2}. In fact, if f ≥ n+k −1 for i ∈ {1,2}, then
i i
n+2k−2≥ |F| ≥ 2n+2k−2, it is impossible. If f = n+k−1 and f = n+k−2, then
1 2
n+2k−2 ≥ |F| ≥ 2n+2k−3, i.e. n ≤ 1 it is impossible because of n ≥ 2.
By Lemma 6, for i ∈ {1,2}, if Di −F is disconnected, then Di −F has two
k−1,n i k−1,n i
components, one of which is a trivial component, say x . Let B = Di − F − {x },
i i k−1,n i i
B is connected to DJ −F by the similar discussion of Case 1. Thus, if D −F is
i k−1,n J k,n
disconnected, then either it has two components, one of which is a trivial component or an
edge; or has three components, two of which are trivial components.
Corollary 1. Let D be the data center network with k ≥ 2 and n ≥ 2. Then the g-
k,n
good neighbor diagnosabilities of D under the PMC model and the MM model are both
k,n
t (D )= (g+1)(k−1)+n+g for 1≤ g ≤ n−1.
g k,n
Proof. By Lemma 3, D is (n+k−1)-regular and (n+k−1)-connected and N = t ≥
k,n k,n
(n+ 1)2k − 1. By Lemma 5, κg(D ) = (g +1)(k −1)+n if 0 ≤ g ≤ n−1 and k ≥ 1.
2 2 k,n
Since N−[2κg(D )+3κ1(D )+2g−n−1]≥ (n+1)2k −1−[(g+1)(k−1)+n+3(n+
k,n k,n 2 2
2k−2)+2g−n−1] = (n+ 1)2k −(g+7)(k−1)−3n−2g+ 1 > 0 for n ≥ 2, k ≥ 2 and
2 2
1 ≤ g ≤n−1, Condition (2) in Theorem 1 holds; By Lemma 4, Condition (1) in Theorem 1
holds; Condition (3) in Theorem 1 holds by Lemma 9. By Theorem 1, the corollary holds.
10