Super Connectivity of Iterated Line Digraphs

Transcription

1 Super Connectivity of Iterated Line Digraphs Ding-Zhu Du Hung Q. Ngo Lu Ruan Jianhua Sun Abstract Many interconnection networks can be constructed with line digraph iterations. In this paper, we will establish a general result on super connectivity based on the line digraph iteration. A digraph has super connectivity if it has connectivity and every node-cut of cardinality consists of either the ending nodes of all non-loop out-edges, that come from a node, or the starting nodes of all non-loop in-edges, that end at a node. Key Words: line digraph iterations, super connectivity, interconnection networks. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455, USA. dzd, hngo, ruan, Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing , China. Support in part by 973 Information Technology and High-Performance Software Program of China. 1

2 1 Introduction Consider a digraph. Clearly, all ending nodes of out-edges at any node form a node-cut. All starting nodes of in-edges at any node also form a node-cut. Those node-cuts are called natural node-cuts. A digraph has super connectivity if it has connectivity and every node-cut of cardinality is natural. The super connectivity is an important issue studied in interconnection networks [2, 6, 18, 22]. The line digraph iteration preserves the degree, that is, the line digraph of a -regular digraph is still -regular. Due to this important property, many interconnection networks are constructed with line digraph iterations [3, 7, 10, 16, 19, 20]. Can the line digraph iteration preseve the super connectivity? In this paper, we address this question by presenting some general results and their applications. 2 Main Result Consider a digraph. The line digraph of is defined by!# %!'&(*)+ that is, takes the edge set of as its node set and there exists an edge from a node, to another node - in if, ends at the starting node of - in. Suppose has super connectivity. Could this imply that. has super connectivity? The answer is NO. A counterexample is as shown in Fig. 1. This situation leads us to study the super edge-connectivity of, which has close connection with the super connectivity of.. An edge-cut is natural if it consists of either all non-loop out-edges at a node or all non-loop in-edges at a node. A digraph has super edge-connectivity if it has edge-connectivity and every edge-cut of cardinality is natural. Note that a natural node-cut of. may not be a natural edge-cut of. Thus, the following result needs a little proof. 2

3 G L(G) Figure 1: A counterexample. Lemma 1. has super edge-connectivity if and only if. has super connectivity. Proof. First, assume has super edge-connectivity. Then has connectivity. Thus, every node has out-degree and in-degree at least. Consider a node-cut of, with cardinality. is an edge-cut of and hence is natural. Without loss of generality, assume that consists of all non-loop out-edges at a node. If has a loop, then the loop % has out-degree and is exactly the set of ending nodes of those out-edges. Hence, is a natural node-cut of. If has no loop, then every in-edge at as a node in has out-degree and is exactly the set of ending nodes of those out-edges. Hence, is a natural node-cut of. Conversely, assume. has super connectivity. Then has edge-connectivity. Consider an edge-cut of, with cardinality. is a node-cut of. and hence is natural in. Without loss of generality, assume that is the set of ending nodes of all out-edges at a node in.. Note that does not contain loop in since is a minimum edge-cut of. Therefore, consists of all out-edges at in, that is, is natural. It is quite interesting that the super edge-connectivity is preserved by line digraph iterations, while the super connectivity is not. Theorem 1. If has super edge-connectivity, then. has super edge-connectivity. Proof. Consider a minimum line-cut of. Suppose breaks the node set of into 3

4 two parts and such that no edge other than those in is from to. Let We next show several claims. Claim 1.. & ) & ) % %! & ) Proof. By Lemma 1, has super connectivity and hence edge-connectivity at least. This means. On the other hand, since has super edge-connectivity, there exists a node of such that either all non-loop out-edges at or all non-loop in-edges at form a minimum edge-cut of, with cardinality. Without loss of generality, assume the former occurs. If has a loop, then this loop has out-degree in. If has no loop, then every in-edge of has out-degree in. This means that in any case, has edge-connectivity at most. Hence,. Claim 2. or Proof. For contradiction, suppose. and, %, & ).. Define where. is the node set of. Note that - -%(& ). Thus, and. Moreover, every edge -, from a node - in to a node, in must belong to (since -, & implies - & ) and hence belongs to. Therefore, is an edge-cut of. Similarly, we can show that is an edge-cut of. Note that and. Since has super edge-connectivity, both and are natural edge-cuts of cardinality. In particular, and do not contain any loop. Hence, we have. It follows that any two edges in cannot share the same ending vertex. Therefore, must consist of out-edges at a node, and must consist of in-edges at a node - (Fig. 2). It also follows that. 4

5 v x y Figure 2:. Choose (&. Note that and do not contain any loop. Then every out-edge at, not in, must belong to. Thus, any path from to - not passing edge -% must pass some edge in, ). (Otherwise, the path will go from an edge in to an edge in. This produces an edge from to in, not in, a contradiction.) This means that, )+ - ) is also an edge-cut of, with cardinality, which is not natural, contradicting the super edgeconnectivity of. Claim 3. is natural. Proof. First, we show %. For contradiction, suppose. Note that each node in has at least non-loop out-edges and at least non-loop in-edges. Moreover, and. Thus, each node in must have an in-edge not in and an out-edge not in. Such an in-edge must belong to that and, contradicting Claim 2. and such an out-edge must belong to Now, assume ). Without loss of generality, assume also. This means. Note that if has a non-loop out-edge not in, then it must belong to. Therefore, all non-loop out-edges at belong to. Since has at least non-loop out-edges and, we have and that is exactly the set of all non-loop out-edges at. It follows that. Hence, if has no loop, then is natural. If has a loop, then this loop must belong to. In fact, the loop being in 5

6 would introduces an edge from to, but not in, and the loop being in would introduces an edge from to, but not in, a contradiction. Thus, the loop not being in implies it being in. Since, can have only one loop. Hence, is exactly the set of all non-loop out-edges at the node % in., that is, is natural. By Claims 1 and 3, every minimum edge-cut of is a natural edge-cut of cardinality. Therefore, has super edge-connectivity. Corollary 1. If has super edge-connectivity, then for has super connectivity. The counterexample in Fig. 1 tells us that in general, a digraph having super connectivity may not have super edge-connectivity. However, Theorem 1 tells us that this is true for a special family of digraphs line digraphs. 3 Applications When an interconnection network contains possible node-faults there are two fault-tolerance measures in the literature. The first one is the connectivity. The second one is the probability of the remaining network being connected when nodes fail with certain probabilistic distribution. Let be the family of all node-cuts of a digraph. For each set of nodes, let denote the probability of all nodes in being faulty. By the inclusion-exclusion principle, connected disconnected # %! ) % When all nodes ) are independent, ( % nodes in (in then ' ) is a product of failure probabilities of ). Therefore, if every node has the same fault probability of a small number, connected depends mainly on the number of the minimum node-cuts. The number 6

7 of the minimum natural node-cuts is certainly a lower bound of the number of minimum node-cuts. Therefore, the super-connected digraph reaches maximum fault-tolerance in certain sense. Given a degree bound, many constructions have been found in the literature to achieve the maximum connectivity and near-minimum diameter [21, 8], including the Kautz digraphs [16], cyclically-modified de Bruijn digraphs [8, 17], generalized cycles [10], etc. Do they also have super connectivity? In this section, we study some of them. Example 1. The Kautz digraph is the complete digraph on vertices without loop and in general [16]. We claim that has super line-connectivity. Consider a line-cut of size in, which breaks the vertex set of into two parts and such that every link from to belongs to. Note that there are links from to and each vertex has out-links. Therefore, line-cut.. That is,. Thus, or. Since implies, is a natural Corollary 2. The Kautz digraph has super connectivity for. Example 2. Ferrero and Padró [10] studied a family of digraphs where * for and is a -partite digraph that some ( and that an edge % exists if and only if & and & ). We claim that for and, * has super lineconnectivity. To show it, consider a line-cut of cardinality at most, which breaks the vertex set into two parts and such that every edge from to belongs to. Denote and. Then we must have First, we show that there exists an such that! or #. For contradiction, suppose such an does not exist. Then for every,!%& and #'. Note that ( #. Therefore, a contradiction. # ) 7 for

8 Now, suppose, without loss of generality, that for some. Since )!&, there exists an such that and (denote ). Without loss of generality, assume and ). Then. This implies and ). This in turn implies!. Hence,. Since contains no loop, consists of out-edges at the node in. This means that is natural. Corollary 3. For,, and, has super connectivity. Example 3. Ferrero and Padró [10] also studied a family of digraphs. Here, is a directed cycle of length, is the generalized Kautz digraph with node set +) and edge set *& ). The operation is defined as follows. Let and, then has vertex set and link set. %& & ). It is not hard to prove that for and that is a -partite digraph with. Moreover, each node in has out-edges which reach consecutive nodes in, the out-edges of those consecutive nodes in reach consecutive nodes in,, that is, take each node in as a root, we can find a complete -nary tree such that the second level consists of consecutive nodes in, the third level consists of consecutive nodes in,..., the -level consists of consecutive nodes in # which are exactly those nodes in other than the root. Now, we claim that has super line-connectivity. To show it, consider a line-cut with cardinality at most, which breaks the node set into two nonempty parts and such that every edge from to belongs to. First, we note that there must exist an such that and and (note: contradicting.. In fact, if such an does not exist, then there must exist ). It follows that contains all edges from to Without loss of generality, we may assume and. Construct a digraph as follows: %(& if and only if there exists a path of length from to in. From the property of described above, it is easy to see that is isomorphic to. Note that if there exists a path of length from to 8,

9 in, then such a path is unique. This means that each edge of uniquely corresponds to a path of length between two nodes in. Let be the set of edges corresponding to those paths containing an edge in. Then is an line-cut of. Since each edge in can be contained in exactly such paths (Fig. 3), we have. Note that has super line- 1 2 d d k connectivity. Therefore, that must break all 1 d p-k-1 Figure 3: paths pass through the same edge. or. First, assume paths which form a complete -nary tree rooted at a node in be done only if consists of out-edges at the root. Hence, is natural. Similarly, also implies that is natural.. This means. This can Corollary 4. For and, has super connectivity. 4 Discussion The line digraph iteration preserves the degree, that is, the line digraph of a -regular digraph is still -regular. This is a very important property different from line graph iteration. This property enable the line digraph iteration to become a very useful tool to construct interconnection networks. In this paper, we showed that the line-digraph iteration preserves the super connectivity under certain condition. We also established that two families of generalized cycles are super connected. Recently, generalized cycles have been studied extensively [4, 5, 10, 12]. They contain many important inter- 9

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