International Journal of Software Engineering and Its Applications Vol.8, No.7 (2014), pp.65-72 http://dx.doi.org/10.14257/ijseia.2014.8.7,06
Constructing Node (Edge) Disjoint Paths on the Petersen Graph Jung-hyun Seo1, Myeongbae Lee2, Jongseok Kim3 and HyeongOk Lee4* 1
Dept. of Computer Eng., National Univ. of Sunchon, 255 Jungang-ro, Sunchon, Chonnam, 540-950, Republic of Korea 2 Dept. of Information and Communication Eng., National Univ. of Sunchon, 255 Jungang-ro, Sunchon, Chonnam, 540-950, Republic of Korea 3 Dept. of Computer Sci., Univ. of Rochester, Rochester, NY 14627, USA 4 Dept. of Computer Edu., National Univ. of Sunchon, 255 Jungang-ro, Sunchon, Chonnam, 540-950, Republic of Korea 1
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[email protected] Abstract In a (d, k)-graph problem, the Petersen graph has a maximum of 10 nodes at degree 3 and diameter 2. Using the Petersen graph, a folded Petersen cube network, a hyper-Petersen network, a cube-connected Petersen network, a cyclic Petersen network, a Petersen-torus network and a 3D Petersen-Torus network have been suggested. In an interconnection network, a disjoint path is an important research topic for efficient and reliable routing. In this study, a disjoint path on the Petersen graph was analyzed so that it could be applied to construction of a disjoint path in a network to be made using the Petersen graph or one to be developed in any other way. It was demonstrated that three-node disjoint paths can be constructed between any two nodes (one-to-one), and that node disjoint paths can be constructed between any one node and any three nodes (one-to-many). The maximum nodes disjoint path in the Petersen graph is three. The length of every disjoint path was not greater than the Petersen graph diameter plus two, and node disjoint paths were edge-disjointed. Keywords: Node disjoint path, Petersen graph, Interconnection network, Parallel path
1. Introduction Over the past years, High Performance Computing (HPC) has been extensively studied by researchers and applied to a variety of field ranging from science, engineering, and business. With the development of HPC, there are many computing paradigms including multicomputer, multiprocessor, Grid computing, Cloud computing, Peer-to-Peer (P2P) computing etc., The role of interconnection network is very important so that the parallel processing system can efficiently perform various application algorithms in engineering and scientific. Various interconnection networks have been announced to date for parallel processing system. According to composition of nodes and edges, static network can be classified into mesh family, hypercube family, and star graph family. Mesh family includes torus, honeycomb mesh, diagonal mesh, hexagonal mesh, etc., hypercube family includes hypercube, folded hypercube, multiply-twisted-cube, etc., and star graph family includes star graph, macro-star, transposition graph, etc., [18]. In interconnection networks (graph theory), degree is the number of edges incident to a certain node. The minimum number of edges between two nodes is called the distance, and the maximum distance within a network is *
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ISSN: 1738-9984 IJSEIA Copyright ⓒ 2014 SERSC
International Journal of Software Engineering and Its Applications Vol.8, No.7 (2014)
called the diameter. Network cost is a product of degree, and diameter and is an important measure for the evaluation of interconnection networks [1]. Finding a graph to minimize degree and diameter in the given number of nodes corresponds to finding an effective multiprocessor. On the other hand, finding a graph having the maximum number of nodes where degree d and diameter k are given is a (d ,k)-graph problem, and such a graph is called a (d, k)-graph [2]. In a (d, k)-graph problem, an upper bound can be obtained by considering a Moore graph. When d and k are given, there is no graph having a higher number of nodes compared to a Moore graph [3]. Figure 1(b) is a (3, 2)-Moore graph. The shape is identical to the Petersen graph in Figure 1(a). The maximum number of nodes given for a graph with d=3, k=2 is 10 [4, 5]. Based on the merits of the network cost of the Petersen graph, various types of network have been suggested [6-10, 16]. Routing, Color problem, Hamilton path, Cycle and Girth, Symmetry, k-Snarks, Diversity, etc., have been published in Petersen graph [19, 20]. In addition, until recently, many studies using the Petersen graph have been published in computer science and mathematics such as [13-17]. In interconnection networks, a disjoint path is used when any two nodes exchange a packet at the same time [11]. It is also used for quick and reliable routing when transmitting a packet from any one node to another node. With many paths offered, packets can be simultaneously transmitted by divisions, and congestion of packets in a communication link can be avoided. In addition, upon a failure, a detour communication link is provided, enabling fault-tolerant routing [12]. For this reason, disjoint paths are very important and are categorized into three types: one-to-one, one-to-many, and many-to-many. Studies of these three methods were listed by Lai, et al., [13]. This study demonstrates that node disjoint paths exist in the Petersen graph, and suggests how to construct three paths of lengths ≤4 for networks using the Petersen graph, which has already been, or is to be, proposed. In this paper, Section 2 describes the attributes of the Petersen graph and demonstrates that node disjoint paths exist between any two nodes. Additionally, several definitions are mentioned in the paper. Section 3 discusses the following: first, how to construct paths between any two nodes is suggested, and we demonstrate that paths constructed by the suggested method are disjointed from each other; and second, how to construct paths between any one node and another three nodes (exclusive of the first node) is suggested, and we demonstrate that paths constructed by the suggested method are disjointed from each other. Finally, our conclusions are presented.
2. Attribute Petersen Graph and Node Disjoint Path The Petersen graph has 10 nodes. In Figure 1(a), there is a pentagonal cycle consisting of 5 nodes on the outside with a star cycle also consisting of 5 nodes on the inside. The nodes of the outer cycle are connected to each node of the inner cycle. The Petersen graph is a regular graph, a node- and an edge-symmetry graph, with degree 3, diameter 2, connectivity 3 and girth 5 [5]. There are several ways to assign Petersen graph node addresses, and the way shown in Figure 1(a) is described by Das and Banerjee [6], Ohring and Das [7] and Seo [8].
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Copyright ⓒ 2014 SERSC
International Journal of Software Engineering and Its Applications Vol.8, No.7 (2014)
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(a) Petersen Graph
(b) (3, 2)-Moore Graph
Figure 1. Petersen Graph and (3, 2)-Moore Graph In the Petersen graph, P=(Vp, Ep). x,y∈{1-5}, xt2, T = t1t2, and if t1
