Structure Connectivity and Substructure Connectivity of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-Ary <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-Cube Networks

We present new results on the fault tolerability of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-ary <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-cube (denoted <inline-formula> <tex-math notation="LaTeX">$Q_{n}^{k}$ </tex-math></inline-formula>) networks. <inline-formula> <tex-math notation="LaTeX">$Q_{n}^{k}$ </tex-math></inline-formula> is a topological model for interconnection networks that has been extensively studied since proposed, and this paper is concerned with the <italic>structure/substructure connectivity</italic> of <inline-formula> <tex-math notation="LaTeX">$Q_{n}^{k}$ </tex-math></inline-formula> networks, for <italic>paths</italic> and <italic>cycles</italic>, two basic yet important network structures. Let <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> be a connected graph and <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> a connected subgraph of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. The <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-<italic>structure connectivity</italic> <inline-formula> <tex-math notation="LaTeX">$\kappa (G; T)$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is the cardinality of a minimum set of subgraphs in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, such that each subgraph is isomorphic to <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>, and the set’s removal disconnects <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. The <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-<italic>substructure connectivity</italic> <inline-formula> <tex-math notation="LaTeX">$\kappa ^{s}(G; T)$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is the cardinality of a minimum set of subgraphs in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, such that each subgraph is isomorphic to a connected subgraph of <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>, and the set’s removal disconnects <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. In this paper, we study <inline-formula> <tex-math notation="LaTeX">$\kappa (Q_{n}^{k}; T)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\kappa ^{s}(Q_{n}^{k}; T)$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$T=P_{i}$ </tex-math></inline-formula>, a path on <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> nodes (resp. <inline-formula> <tex-math notation="LaTeX">$T=C_{i}$ </tex-math></inline-formula>, a cycle on <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> nodes). Lv <italic>et al.</italic> determined <inline-formula> <tex-math notation="LaTeX">$\kappa (Q_{n}^{k}; T)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\kappa ^{s}(Q_{n}^{k}; T)$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$T\in \{P_{1},P_{2},P_{3}\}$ </tex-math></inline-formula>. Our results generalize the preceding results by determining <inline-formula> <tex-math notation="LaTeX">$\kappa (Q_{n}^{k}; P_{i})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\kappa ^{s}(Q_{n}^{k}; P_{i})$ </tex-math></inline-formula>. In addition, we have also established <inline-formula> <tex-math notation="LaTeX">$\kappa (Q_{n}^{k}; C_{i})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\kappa ^{s}(Q_{n}^{k}; C_{i})$ </tex-math></inline-formula>.

is a topological model for interconnection networks that has been extensively studied since proposed, and this paper is concerned with the structure/substructure connectivity of Q k n networks, for paths and cycles, two basic yet important network structures.Let G be a connected graph and T a connected subgraph of G.
The T -structure connectivity κ(G; T ) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to T , and the set's removal disconnects G.The T -substructure connectivity κ s (G; T ) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to a connected subgraph of T , and the set's removal disconnects G.In this paper, we study κ(Q k n ; T ) and κ s (Q k n ; T ) for T = P i , a path on i nodes (resp.T = C i , a cycle on i nodes).In [11], κ(Q k n ; T ) and κ s (Q k n ; T ) were determined for T ∈ {P 1 , P 2 , P 3 }.Our results generalize the preceding results by determining κ(Q k n ; P i ) and κ s (Q k n ; P i ).In addition, we have also established κ(Q k n ; C i ) and κ s (Q k n ; C i ).

I. INTRODUCTION
I NTERCONNECTION networks play an important role in large-scale multiprocessor systems.Like most networks, an interconnection network can be represented by a graph G = (V (G), E(G)), where nodes in V (G) correspond to processors, and edges in E(G) correspond to communication links.

A. CONNECTIVITY OF INTERCONNECTION NETWORKS
The fault tolerance of interconnection networks has always been an important issue.One crucial parameter to evaluate the fault tolerability of a network is its connectivity.The connectivity of a graph G, denoted by κ(G), is the minimum cardinality of a node set F ⊆ V (G), such that F 's deletion disconnects G.As variants of the classic node-connectivity, several kinds of conditional connectivity were proposed and studied [2], [3], [5], [6], [8], [9], [12]- [16], [18], [23], [25], [26].Notably among them, Fàbrega and Fiol [4] introduced the g-extra connectivity.The g-extra connectivity κ g (G) of a connected graph G is the minimum cardinality of a set of nodes in G , if such a set exists, whose deletion disconnects G and leaves each remaining component with at least g + 1 nodes.Obviously, κ 0 (G) = κ(G), making κ g (G) a generalization of κ(G).
Lin et al. [17] considered the fault status of a certain structure, rather than individual nodes, and proposed structure connectivity and substructure connectivity.Let G be a connected graph, and T a connected subgraph of G.The T -structure connectivity κ(G; T ) of G is the cardinality of a minimum set of subgraphs F = {T 1 , T 2 , . . ., T m } in G, such that every T i ∈ F is isomorphic to T , and F 's deletion disconnects G.The T -substructure connectivity κ s (G; T ) of G is the cardinality of a minimum set of subgraphs F = {H 1 , H 2 , . . ., H m }, such that every H i ∈ F is isomorphic to a connected subgraph of T , and F 's deletion disconnects G.By definition, κ(G; T ) ≥ κ s (G; T ).The structure connectivity and substructure connectivity have been studied for some well-known networks [11], [17], [21], [22], [27].

B. APPLICATIONS OF STRUCTURE/SUBSTRUCTURE CONNECTIVITY AND OUR CONTRIBUTIONS
The traditional connectivity assumes that the status of a node is an event independent of the status of nodes around it.However in reality, nodes that are linked could affect each other, and the neighbours of a faulty node are more likely to fail than other nodes.Moreover, in the NoC technology (Network-on-Chip), part or whole of a network of interest are made on a chip, which means that the failure of any node on the chip can be considered the failure of the whole chip.All these motivated the research on fault tolerance of networks based on some certain structures rather than individual nodes.The study of structure fault tolerance is therefore of both scientific value and practical significance.
In this paper, we focus on two basic structures of all networks: paths and cycles.Let P i be a path on i nodes, and C i a cycle on i nodes, respectively (see Fig. 1).Paths and cycles in a network are very important structures, both for basic network functionality and for implementing many algorithms executing on networks.When nodes in a path or cycle become faulty, the impacted path/cycle cannot function as a whole.So the whole path/cycle can be viewed as faulty.In many cases, it is easier to identify and locate a faulty structure than individual nodes in the structure.There are already many works in the literature studying path/cyclestructure fault tolerance for some well-known networks.For example, Lin et al. [17] investigated {P 2 , P 3 , C 4 }structure/substructure connectivity for hypercubes.Wang et al. [22] established {C 3 , C 4 }-structure/substructure connectivity for generalized hypercubes.The general {P i , C i }structure/substructure connectivity have been studied for hypercubes, folded hypercubes and bubble-sort graphs [21], [27].In this paper, we determine the path-and cyclestructure/substructure connectivity for k-ary n-cubes.The newfound results further our understanding of k-ary n-cubes, and furnish more parameters to consider when evaluating and selecting an interconnection network.Lv et al. [11] The results in this paper are summarized as follows.
For Q 3 n , we have: We have: Of particular note is that a definitive structure connectivity for odd-cycles in Q k n still remains elusive.Our result of κ(Q k n ; C 2l+1 ) ≤ 2n − 2 provides an upper-bound on the structure connectivity for odd-cycles.This "halfsolved" κ(Q k n ; C 2l+1 ) and the unknown κ(Q 3 n ; C 3l+1 ) are the two missing pieces for a complete solution to Q k n 's structure/substructure connectivity for paths and cycles.
The rest of this paper is organized as follows.In Section 2, we introduce definitions and notations used throughout the paper.Section 3 establishes κ(Q 3 n ; T ) and κ s (Q 3 n ; T ) for T ∈ {P i , C i }.In Section 4, we determine κ(Q k n ; T ) and κ s (Q k n ; T ) for k ≥ 4, T ∈ {P i , C i }.Section 5 concludes the paper.

II. PRELIMINARIES
The k-ary n-cube Q k n is a popular interconnection network for parallel systems which has been proved to possess many attractive properties such as regularity, node transitivity and link transitivity.A number of parallel systems have been built with a k-ary n-cube forming the underlying topology, such as the J-machine [19], the iWarp [20] and the Cray T3D [10].In particular, the 3-ary n-cube Q 3 n has been widely deployed in interconnections of parallel systems like the IBM Blue Gene/Q [1].
The k-ary n-cube Q k n (k ≥ 2 and n ≥ 1) is a graph consisting of k n nodes, each of which has the form u = a 1 a 2 . . .a n , where 0 ≤ a i ≤ k − 1 for 1 ≤ i ≤ n.Two nodes u = a 1 a 2 . . .a n and v = b 1 b 2 . . .b n are adjacent if and only if there exists an integer j, 1 ≤ j ≤ n, such that a j = b j ± 1 (mod k) and a i = b i , for every i ∈ {1, 2, . . ., n} \ {j}.Such a link uv is called a j-dimensional link.For clarity of presentation, we omit writing "(mod k)" in similar expressions for the remainder of the paper.Note that each node has degree 2n when k ≥ 3, and n when k = 2. Obviously, Two distinct adjacent nodes are neighbours.The set of neighbours of a node v in a graph G is denoted by n by deleting the nodes of F i (resp.F i ) together with their incident links.
The following lemmas are useful in Sections 3 and 4.

III. THE STRUCTURE CONNECTIVITY AND SUBSTRUCTURE CONNECTIVITY OF Q 3 n
In this section, we determine κ(Q 1) . . .a n , and let u +,− i,j = a 1 . . .(a i + 1) . . .(a j − 1) . . .a n .Similarly, u +,+ i,j , u −,− i,j , u −,+ i,j can be defined.Let Then P (u) is a path and is called the neighbour structure of u (see Fig. 3).Let is a path lying on P (u) for 1 ≤ i, j ≤ n.For convenience, we denote such a path P by Similarly, consider the neighbour structure of v.It is easy to see that the neighbour structure of u and v has exactly two common nodes u The neighbour structure of u and v.
] with p odd, and 3l+1 lying on P (u) and 3l+2 lying on P (u) and To make the number of faulty subgraphs of C 3l+s minimum which contain the nodes in N (x), we should construct as many P 3l+s 's/C 3l+s 's as possible and each P 3l+s /C 3l+s need to contain as many nodes in N (x) as possible.By Lemma 5, Q 3 n contains no structure A, and so any three nodes in N (x) are not three consecutive nodes on a path/cycle.Combining this with the definition of the neighbour structure of x, each P 3l+s /C 3l+s contain at most 2l + s nodes in N (x).Note that Combining this with Lemma 6, we have κ s (Q 3 n ; P 3l+s ) ≥ 2n 2l+s for n ≥ 2 and l ≥ 0. Recall that κ(Q 3 n ; P 3l+s ) ≥ κ s (Q 3 n ; P 3l+s ).Lemma 3 yields the following result.
For C 3l , we will successively find 3l is indeed a cycle on 3l nodes.Next assume that q ≥ 2.
If q is odd, then assume that q = 2s + 1.First suppose that s = 0.
To make the number of faulty C 3l+s 's minimum which contain the nodes in N (x), each C 3l+2 need to contain as many nodes in N (x) as possible.By Lemma 5, Q 3 n contains no structure A, and so any three nodes in N (x) are not three consecutive nodes on a cycle.Combining this with the definition of the neighbour structure of x, a cycle C 3l+2 contain at most 2l + 1 nodes in N (x).Note that |N (x)| = 2n.

IV. THE STRUCTURE CONNECTIVITY AND SUBSTRUCTURE CONNECTIVITY OF Q k n
In this section, we determine κ(Q k n ; T ) and = a 1 . . .(a j − 1)(a j+1 − 1) . . .a n .Similarly, u +,− j,j+1 and u +,+ j,j+1 can be defined.Let with n even (see Fig. 7), and with n odd (see Fig. 8).Then P (u) is a path and is called the neighbour structure of u.
Similarly, consider the neighbour structure of v.It is easy to see that the neighbour structure of u and v has exactly two common nodes u + n = v − 1 and u + 1 = v − n (see Fig. 7 and Fig. 8).For convenience, no matter what the parity of n is, the above neighbour structure of u and v is denoted by 2n with u 2n = v 1 (see Fig. 9).Let P = u i u i u i+1 u i+1 u i+2 . . .u j−1 u j−1 u j be a path lying on P (u) for 1 ≤ i, j ≤ 2n.For convenience, we denote such a path P by [u i , u j ].Similarly, [u i , u j ] can be defined for 1 ≤ i, j ≤ 2n − 1.
n Fig. 7.The neighbour structure of u and v with n even.
Fig. 8.The neighbour structure of u and v with n odd.Fig. 9.The neighbour structure of u and v.
Lv et al. [11] proved the following theorem about κ(Q k n ; P i ) and κ s (Q k n ; P i ) for i = 1, 3. In this subsection, we generalize the theorem by establishing κ(Q k n ; P i ) and Consider the neighbour structure P (u) and P (v) of u and v.
To make the number of faulty subgraphs of C 2l minimum which contain the nodes in N (x), we should construct as many P 2l 's/C 2l 's as possible and each P 2l /C 2l need to contain as many nodes in N (x) as possible.Since Q k n contains no triangles for k ≥ 4, any two nodes in N (x) are not two consecutive nodes on a path/cycle.Combining this with the definition of the neighbour structure of x, each P 2l /C 2l contain at most l nodes in N (x).Note that Set 2l + 1 = 1, 3 in the Theorem 6.Then Theorem 5 given by Lv et al. in [11] is an immediate corollary of Theorem 6.
In this subsection, we investigate the cycle-structure/substructure connectivity for Q k n .
Consider the neighbour structure P (u) and P (v) of u and v.Note that P (u) = u 1 u 1 u 2 u 2 u 3 . . .u 2n−1 u 2n−1 u 2n .In the following, we first give a claim which can be used to construct the desired cycles.
Claim 1.For any u i , u j ∈ V (P (u)) with 1 ≤ i < j ≤ 2n and j − i ≥ 3, there exists u By the definition of u i , we have u Without loss of generality, assume that u i = u − s and u j = u + t .By the definition of P (u) and j − i ≥ 3, we have s < t.Let u i,j = u −,+ s,t .Then u i,j ∈ V (P (u)) and u i u i,j u j is a P 3 .The claim holds.
We will successively find Next consider l < 2n and assume that 2n = pl + q.Then p + 1 = 2n l .By Fig. 12.An example of C k , C 1 k+2s and C 2 k+2s in Q k 2 .
q q q q q q q q q q q q q q q q q q q q q q . . .

V. CONCLUSION
In a given network, how many of a particular structure can go faulty, and the network still remains connected?That is the question this paper tried to address.It established structure connectivity κ(Q k n ; T ) and substructure connectivity κ s (Q k n ; T ), where k ≥ 3, and T is a path or cycle, both being basic yet important structures in all computer networks.Our work not only generalized the known result on path structures [11], but also extended it to cycle structures.These results reveal new characteristics of Q k n , affording more insights into this important network.) is yet to be determined; and (2) The paper's result on structure connectivity for odd-node cycles, κ(Q k n ; C 2l+1 ) with odd k ≥ 5, is an upper-bound, instead of a definitive connectivity.These two sub-problems proved to be challenging, and solving them will completely solve the Q k n 's structure/substructure connectivity for paths and cycles.New and more innovative approaches, different than ours used in this paper, might be in order.

Fig. 1 .
Fig. 1.A path P i and a cycle C i .

Lemmas 7 and 8
yield the following result.

G.l+1 − 1 .
Zhang and D. Wang: Structure Connectivity and Substructure Connectivity of k-Ary n-Cube Networks 2n In order to prove that κ s