Super edge-connectivity and matching preclusion of data center networks

07/13/2018 ∙ by Huazhong Lü, et al. ∙ NetEase, Inc 0

Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. k-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network D_k,n is super-λ for k≥2 and n≥2, super-λ_2 for k≥3 and n≥2, or k=2 and n=2, and super-λ_3 for k≥4 and n≥3. Moreover, as an application of k-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of D_k,n. In particular, we have shown that D_1,n is isomorphic to the (n,k)-star graph S_n+1,2 for n≥2.

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1 Introduction

Let be a graph, where is the vertex-set of and is the edge-set of . The number of vertices of is denoted by . The degree of a vertex in is denoted by . For any , we use to denote the subgraph of induced by . For other standard graph notations not defined here please refer to [3].

Networks are usually modeled as graphs, and the edge-connectivity is a classic measurement for the fault tolerance of the graph. In general, the larger the edge-connectivity of the graphs, the higher the reliability of the corresponding networks. It is well-known that , where and are the edge-connectivity and the minimum degree of , respectively. To precisely measure the reliability of graphs, Esfahanian and Hakimi [20] introduced a more refined index, namely the restricted edge-connectivity. Later, Fàbrega and Fiol [21] introduced the -restricted edge-connectivity as a generalisation of this concept.

An edge-cut is called a -restricted edge-cut if every component of contains at least vertices (). The -restricted edge-connectivity , if exists, is the minimum cardinality over all -restricted edge-cuts in . Let be a vertex subset of and let be the complement of , namely . We denote the edges between and by . The minimum -edge degree of a graph for integers , is

and is connected.

For a graph satisfying , if holds, then it is called -optimal. In particular, is the restricted edge-connectivity, and accordingly is known as the edge degree.

For , it was shown that each connected graph of order at least 4 except a star () has a restricted edge-cut and satisfies [20]. Moreover, Bonsma et al. [4] have shown that if exists, then .

A graph is super- (resp. super-) if each minimum edge-cut (resp. -restricted edge-cut) isolates a singleton (resp. a connected subgraph of order ). It is obvious that if is super-, then is -optimal, whereas the reverse does not hold. Generally, a graph is super -edge-connected of order if when at least edges deleted, the resulting graph is either connected or it has one big component and a number of small components with at most vertices in total. Obviously, a super- graph is super -edge-connected of order 1.

A perfect matching of a graph is an independent edge set that saturates all vertices of . For an edge subset of an graph with even order, if has no perfect matching in , then is called a matching preclusion set of . The matching preclusion number, denoted by mp, is defined to be the minimum cardinality among all matching preclusion sets. Any such set of size mp is called an optimal matching preclusion set (or optimal solution). This concept was proposed by Brigham et al. [6] as a measure of robustness of networks, as well as a theoretical connection with conditional connectivity and “changing and unchanging of invariants”. Therefore, networks of larger mp signify higher fault tolerance under edge failure assumption.

It is obvious that the edges incident to a common vertex form a matching preclusion set. Any such set is called a trivial solution. Therefore, mp is no greater than . A graph is super matched if mp and each optimal solution is trivial. In the random link failure scenario, the possibility of simultaneous failure of links in a trivial solution is very small. Motivated by this, Cheng et al. [9] introduced the following definition to seek obstruction sets excluding those induced by a single vertex. The conditional matching preclusion number of , denoted by mp, is the minimum number of edges whose deletion results in the graph with neither a perfect matching nor an isolated vertex. If the resulting graph has no isolated vertices after edge deletion, a path , where the degree of both and are 1, is a basic obstruction to perfect matchings. So to generate such an obstruction set, one can pick any path in the original graph, and delete all the edges incident to and but not . We define

and are ends of a path of length 2,

where if and 0 otherwise.

Proposition 1.1

.[9] For a graph of even order and , mp holds.

A conditional matching preclusion set of that achieves mp, a set of edges whose removal leaves the subgraph without perfect matchings and with no isolated vertices, is called an optimal conditional matching preclusion set (or optimal conditional solution). An optimal conditional solution of the basic form induced by a 2-path giving is a trivial optimal conditional solution. As mentioned earlier, the matching preclusion number measures the robustness of the requirement in the link failure scenario, so it is desirable for an interconnection network to be super matched. Analogously, it is desirable to have the property that all the optimal conditional solutions are trivial as well. The interconnection network possesses the above property is called conditionally super matched.

Until now, the matching preclusion number of numerous networks were calculated and the corresponding optimal solutions were obtained, such as the complete graph, the complete bipartite graph and the hypercube[6], Cayley graphs generated by 2-trees and hyper Petersen networks [10], Cayley graphs generalized by transpositions and -star graphs [11], restricted HL-graphs and recursive circulant [31], tori and related Cartesian products [12], -bubble-sort graphs [13], balanced hypercubes [27], burnt pancake graphs [22], -ary -cubes [35], cube-connected cycles [25], vertex-transitive graphs [24], -dimensional torus [23], binary de Bruijn graphs [26] and -grid graphs [17]. For the conditional matching preclusion problem, it is solved for the complete graph, the complete bipartite graph and the hypercube [6], arrangement graphs [14], alternating group graphs and split-stars [15], Cayley graphs generated by 2-trees and the hyper Petersen networks [10], Cayley graphs generalized by transpositions and -star graphs [11], burnt pancake graphs [8, 22], balanced hypercubes [27], restricted HL-graphs and recursive circulant [31], -ary -cubes [35], hypercube-like graphs [32] and cube-connected cycles [25]. Particularly, Lü et al. [28] has proved recently that it is NP-complete to determine the matching preclusion number and conditional matching preclusion number of a connected bipartite graph.

Data centers are crucial to the business of companies such as Amazon, Google and Microsoft. Data centers with large number of servers were built to offer desirable on-line applications such as web search, email, cloud storage, on-line gaming, etc. Data center networks , DCell in short, was introduced by Guo et al. [19] for parallel computing systems, which has numerous desirable features for data center networking. In DCell, a large number of servers are connected by high-speed links and switches, providing much higher network capacity compared with the tree-based systems. Several attractive properties of DCell has been explored recently, such as Hamilton property [36], pessimistic diagnosability [18], the restricted -connectivity [37] and disjoint path covers [38].

The restricted edge-connectivity and extra (edge) connectivity of lots of famous networks were studied in [4, 7, 21, 30, 39]. In [37], the authors obtained the restricted -connectivity of the DCell, which is the connectivity of under the restriction that each fault-free vertex has at least fault-free neighbors in . In the same paper, the authors proposed an interesting problem that whether similar results of restricted edge-connectivity apply to the DCell network. In this paper, we study this problem and show that the DCell network is super- for and , super- for and or and , and super- for and . As a direct application of the above result, we obtain the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of the DCell.

The rest of this paper is organized as follows. The definition of the DCell and some notations are given in Section 2. The restricted edge-connectivity of the DCell is computed in Section 3. The (conditional) matching preclusion number of the DCell is obtained in Section 4.

2 Preliminaries

We begin with the definition of the DCell.

Definition 1

.[37] A level DCell for each and some global constant , denoted by , is recursively defined as follows. Let be the complete graph and let be the number of vertices in . For , is constructed from disjoint copies of , where denotes the th copy. Each pair of and () is joined by a unique level edge below.

A vertex of is labeled by , where and . The suffix , of a vertex , has the unique , given by . The vertex of is connected to of .

By the definition above, it is obvious that is the complete graph () and is a 6-cycle. We illustrate some with small parameters and in Fig. 1. By Definition 1, we know that there exists exactly one edge, called a level edge, between and . For convenience, let denote the set of all level edges of . Let and , we denote and for . We use to denote the level edge incident with and to denote its level neighbor.

Fig. 1: Some small DCells.

3 Super edge-connectivity of DCell

It is not hard to see that . Observe that the edges coming from a complete subgraph form a non-trivial minimum edge-cut of , so is not super- for . For and , we have the following result.

Theorem 3.1

. is super- for all and .

Proof. By Definition 1, can be split into copies of , denoted by , . It is clear that every vertex in has exactly one neighbor not in . In addition, there is exactly one edge between and for . Let be any minimum edge-cut of , then . Assume that is disconnected. We need to show that is the set of edges incident to a unique vertex.

Case 1. for . Obviously, is connected since is edge-connected. By contracting each of into a singleton, we obtain a complete graph . Moreover, the edges of obtained above correspond to all level edges in . It is clear that whenever and , therefore, is connected when we delete at most edges. (This fact will be used time and time again in the remainder of this paper.) This implies that is connected, a contradiction.

Case 2. for some . Suppose without loss of generality that . If , then for . Since each vertex in has exactly one neighbor not in , is connected, a contradiction. We now assume that . If is connected, by the discussion in Case 1, then is connected. So we assume that is disconnected and is one of its components. Clearly, is connected. If is a singleton and, furthermore, the level edge incident to is contained in , then is a super edge-cut of ; otherwise is connected. If consists of at least two vertices, noting each vertex of has a neighbor not in , then is connected to , yielding that is connected, a contradiction. Hence, the statement holds. ∎

As mentioned earlier, there exists a non-trivial restricted edge-cut if , which implies that is not super- for all .

Lemma 3.2

.[34] The complete graph is super- for .

Lemma 3.3

. Let be any edge in for and . If is a level 0 edge, then and have exactly common neighbors; if is a level edge, , then and have no common neighbors.

Proof. If is a level 0 edge, then lies in a complete subgraph () of . Clearly, and have exactly common neighbors in this . If and have another common neighbor outside this , then a triangle occurs, which is impossible according to Definition 1. If is a level edge, then and have no common neighbors. This completes the proof. ∎

Theorem 3.4

. for all and .

Proof. Since is ()-regular, we have . Additionally, is not a star for and , then . We only need to show that .

Let be any subset of edges in such that and there is no isolated vertex in . We shall prove that is connected. We may assume that . Suppose without loss of generality that is the largest one among . Then for each . Since , each of is connected. By contracting each into a singleton, we obtain a complete graph . Note that , we have whenever and , which implies that is connected. It remains to show that any vertex in is connected to a vertex in via a fault-free path. If is connected, then is connected. We assume that is disconnected. Thus, .

Suppose that is an arbitrary vertex in . If , we are done. So we assume that . Since there exists no isolated vertex in , there exist an edge incident with () in such that . Moreover, if , we are done. So we assume that . We consider the following two cases.

Case 1. is a level edge, . By Lemma 3.3, and have no common neighbors. Let and , then . It is not difficult to see that there are edge disjoint paths from (resp. ) to a vertex in via (resp. ). Observe that , so is connected to a vertex in via a fault-free path.

Case 2. is a level 0 edge. By Lemma 3.3, and have exactly common neighbors in . In fact, are in a complete subgraph of . Besides, (resp. ) has distinct neighbors (resp. ) outside , . So there exist edge disjoint paths (resp. ) from (resp. ) to (resp. ), where (resp. ) is the level neighbor of (resp. ). If at least one of and is fault-free in , we are done. So we assume that each of and has at least one edge in .

There are at most edges of in the of that contains . Clearly, we only need to consider since when . Clearly, is connected since when . In addition, by Lemma 3.2, is super- when . In other words, when . If is connected, for each , , there are distinct neighbors not in and exactly one level neighbor. Since whenever and , there exists a fault-free path from to a vertex in . If is disconnected, it follows that and there exists a singleton, say , in . Then there are edge disjoint paths from the large component of to . Since whenever , the result follows.

By above, we have shown that is connected, which implies that . Thus, the lemma follows.∎

However, is not super- when since the edges coming from a complete subgraph , namely , form a non-trivial minimum restricted edge-cut.

Lemma 3.5

. is super-.

Proof. Let be any edge subset of with . We shall show that if contains no isolated vertex, then either is connected or isolates an edge of . Notice that is constructed from seven disjoint 6-cycles (), for convenience, denoted by , . We may assume that is the largest one among . We consider the following cases.

Case 1. . It is obvious that each is connected. By a similar argument of Case 1 in Theorem 3.1, it can be shown that is connected.

Case 2. . If for each , then is connected. If contains a singleton , then must be connected to . So we assume that contains an isolated edge . Furthermore, if one of the level 2 edges of and is not in , then is connected to ; otherwise, isolates in . For each component of containing at least three vertices, clearly, it is connected to . So we assume that for some , say . Since and are both 6-cycles, and have at most two components, respectively. Clearly, and is connected. If and are two singletons of and , respectively, and , then isolates in ; otherwise, is connected.

Case 3. . Clearly, is connected. If contains a singleton , then is connected to since contains no isolated vertex. If is a component of with , then is connected to . Thus, is connected.

Obviously, is connected. ∎

Theorem 3.6

. is super- for and , or and .

Proof. Let be any edge subset of with . We keep the notation introduced in Theorem 3.4. By Lemma 3.5, it suffices to consider and . We shall show that if contains no isolated vertex, then either is connected or isolates an edge of . If each of is connected for , then is connected. So we assume that one of is disconnected, say . We consider the following cases.

Case 1. . Clearly, . Furthermore, if each of , , is connected, then is connected. By Theorem 3.1, isolates a singleton of . Since there exists no isolated vertex in , must connect to a vertex in . It is not hard to see that there exists a vertex of the larger part of connecting to a vertex in . Thus, is connected.

Now we may assume that is disconnected. At this time, is connected. Again, (resp. ) isolates a singleton (resp. ) of (resp. ). If , then isolates an edge in ; otherwise, is connected.

Case 2. . Clearly, . Then is connected. Since whenever , by Theorems 3.1 and 3.4, isolates exactly one singleton of when . Hence contains two components and , where is the singleton . Since there is no isolated vertex in , is connected to a vertex in . Obviously, there exists a vertex of connecting to a vertex in . Therefore, is connected.

Note that when and , we have already considered this case in Case 1. Note also that when and , may isolate a singleton or an isolated edge of since . As mentioned earlier, we only consider that isolates an isolated edge, say , of . If and hold, then isolates an isolated edge of ; otherwise, is connected.

Case 3. . It suffices to consider with since when and . If is an isolated vertex in , then the level edge , which implies that is connected to a vertex in . So we assume that is an isolated edge in . If the level edges or , then or is connected to a vertex in ; otherwise, isolates an edge of . For any component of with , noting that at most two edges are deleted outside , each vertex in has a neighbor in . It implies that is connected. Thus, the theorem follows. ∎

In what follows, we shall consider 3-restricted edge-connectivity of . The following lemma is needed.

Lemma 3.7

.[1] The complete graph is super- for .

Theorem 3.8

. for all and .

Proof. Pick out a path of length two or a triangle of for and . Clearly, . It suffices to prove that .

Let with such that there are neither isolated vertices nor isolated edges in . Our objective is to show that is connected. Observe that and is () edge-connected, then at most two of , , are disconnected. In fact, when and , by Theorem 3.6, it implies that is connected. So we assume that and , or and . We consider the following cases.

Case 1. For each , is connected. Since , we have whenever and , by the proof of the Case 1 of Theorem 3.1, is connected.

Case 2. Exactly one of is disconnected. We may assume that is disconnected. Then . Since each of () is connected, we can obtain that is connected. We need the following claim.

Claim. Each vertex in is connected to a vertex in via a fault-free path in .

Proof of the Claim. Let be an arbitrary vertex in . If , we are done. So we assume that . Since there are no isolated vertices in , there is an edge such that . If , we are done. Similarly, we assume that . Moreover, there are no isolated edges in , then there is an edge or , say , in . Again, if , we are done. So we assume that . We consider the following three conditions.

(1) Both of and are level 0 edges. That is, and are vertices of some in . In addition, (resp. , ) has distinct neighbors (resp. , ) in but outside , . So there exist edge disjoint paths (resp. , ) from (resp. , ) to (resp. , ), where (resp. , ) is a level neighbor of (resp. , ). If at least one of , and is fault-free in , we are done. So we assume that each of , and has at least one edge in .

There are at most edges of in the . Since when , we need only to consider . In addition, by Lemmas 3.2 and 3.7, is super- and super- when and , respectively. In other words, when and when . If the containing is connected, for each vertex of (not , and ), , there are edge disjoint paths from to a vertex in . Since whenever and , there exists a fault-free path from to a vertex in , we are done. Now we assume that is disconnected. Since when , we only need to consider .

There is exactly one singleton in since when . It is not difficult to see the claim holds. So we assume that . When at most are deleted from , the resulting graph is either connected, or contains exactly two components, one of which is a singleton or an edge, or contains exactly three components, two of which are singletons. Let the component of containing , and be , then contains at least vertices except , and . There are at least edges to separate from . Note when . In fact, when . When , if we take on the right side, the left side is . Therefore, there exists a fault-free path from to a vertex in .

(2) Either or is a level 0 edge, but not both. Without loss of generality, suppose that is a level 0 edge. Similarly, has distinct neighbors in but outside , and (resp. ) has distinct neighbors (resp. ) in but outside , where , , and . So there exist edge disjoint paths from to , and there exist edge disjoint paths (resp. ) from to (resp. to ). If at least one of , and is fault-free in , we are done. So we assume that each of , and has at least one edge in .

For convenience, we denote the containing and by and the containing by . Thus, there are at most edges of in the and . If one of and , say