An Edge-Isoperimetric Problem for Powers of the Petersen Graph

Bezrukov, Sergei L.; Das, Sajal K.; Elsässer, Robert

doi:10.1007/s000260050003

Cited by 67 publications

(36 citation statements)

References 13 publications

(21 reference statements)

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“…The following two versions of the edge isoperimetric problem of a graph G(V, E) have been considered in the literature [23], and are N P-complete [24]. Problem 1: Find a subset of vertices of a given graph, such that the edge cut separating this subset from its complement has minimal size among all subsets of the same cardinality.…”

Section: Isoperimetric Problemmentioning

confidence: 99%

“…, n, we consider the problem of finding a subset A of vertices of G such that |A| = m and |θ G (A)| = θ G (m). Such subsets are called optimal [23,25]. Further, if a subset of vertices is optimal with respect to Problem 1, then its complement is also an optimal set.…”

Section: Isoperimetric Problemmentioning

confidence: 99%

“…But, it is not true for Problem 2 in general. However for regular graphs a subset of vertices S is optimal with respect to Problem 1 if and only if S is optimal for Problem 2 [23]. In the literature, Problem 2 is defined as the maximum subgraph problem [24].…”

Section: Isoperimetric Problemmentioning

confidence: 99%

See 2 more Smart Citations

Embedding of Recursive Circulants into Certain Necklace Graphs

Rajan

Parthiban

Rajalaxmi³

2015

Math.Comput.Sci.

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Section: Isoperimetric Problemmentioning

confidence: 99%

Section: Isoperimetric Problemmentioning

confidence: 99%

See 1 more Smart Citation

Embedding of Recursive Circulants into Certain Necklace Graphs

Rajan

Parthiban

Rajalaxmi³

2015

Math.Comput.Sci.

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show abstract

“…Clearly, if a subset of vertices is optimal with respect to Problem 1, then its complement is also an optimal set. However, it is not true for Problem 2 in general, although this is indeed the case if the graph is regular [4]. In the literature, Problem 2 is defined as the maximum subgraph problem.…”

mentioning

confidence: 98%

“…Now, we consider another interesting N P-complete problem namely the edge isoperimetric problem [7] which will be used to solve the wirelength problem [8]. The following two versions of the edge isoperimetric problem of a graph G(V, E) have been considered in the literature [4].…”

mentioning

confidence: 99%

Linear Wirelength of Folded Hypercubes

Rajasingh¹,

Arockiaraj²

2011

Math.Comput.Sci.

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Manuel et al. (Discret. Appl. Math. 157 (7): [1486][1487][1488][1489][1490][1491][1492][1493][1494][1495] 2009) obtained the exact wirelength of an r -dimensional hypercube into a path as well as a 2 r/2 × 2 r/2 grid and conjectured the same for a folded hypercube. In this paper we solve the edge isoperimetric problem for folded hypercubes and thereby obtain the exact wirelength of folded hypercubes into paths. Further we compute the exact wirelength of the r -dimensional folded hypercubes into 2 k × 2 r −k grids.

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Reconfigurable Fault‐tolerance mapping of ternary N‐cubes onto chips

Fan

Han

et al. 2020

Concurrency and Computation

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Network-on-chip (NoC) is a new design method of system-on-chip used in very large scale integrated circuit (VLSI) systems. It is an important issue for choosing the appropriate topology for NoC. Wirelength and layout area are significant parameters affecting NoC due to the restriction of chip area. In this paper, we propose a new interconnection network called the incomplete ternary n-cube for parallel computing systems. Then, a linear algorithm is proposed to layout incomplete ternary n-cube network onto torus NoC. Furthermore, the failure of interconnection network is also taken into account, and a fault-tolerant layout of incomplete ternary n-cube with faulty edges into torus NoC is verified. Theoretical analysis demonstrates that the proposed algorithm can reduce the network cost and wirelength, which be conducive to estimate the wire length and chip area. KEYWORDSfault tolerance, graph embedding, network-on-Chip, reconfigurable, wirelength INTRODUCTIONWith semiconductor technology entering the nanometer stage, it has become a reality for the integration of 100 billion transistors in a single chip. How to utilize a large number of transistors effectively is an important issue of the chip architecture. It has become very difficult to improve the overall performance of the system by improving the performance of the single processor core; the difficulty and complexity of chip design are also increasing. 1,2 Network-on-chip (NoC) has the advantages of high integration, low power consumption, low cost, and small volume, and it has become one of the mainstreams of very large scale integration (VLSI) system design. 3Network topology research is an important aspect of NoC. The network on chip uses the topological structure of computer network for reference. Ring-based and torus-based NoC architectures are illustrated in Figure 1 and Figure 2. Due to its own characteristics of NoC, such as a large number of available communication wire resources, a large amount of communication can be carried out between the cores, restricted by the area of the chip, and the layout of the regular network. 4,5 Therefore, NoC has certain special requirements for the topology structure.Due to the restriction of the chip area, the interconnection network and the total wire length become the crucial issues that affect the NoC communications. It is an auxiliary consideration for the cost of NoCs with respect to their interconnection networks.The more complex of networks, the higher in connectivity wiring and costs. Therefore, it is desirable to substitute a simple network for NoC with complex networks as an alternative. In such cases, one of the primary targets is improving network performance by providing better static topology features such as diameter and average distance between vertices. 6-8 Nevertheless, when designing network architecture, it is crucial to consider the influence of physical design constraints, for example wirelength, wiring congestion, and network cost. Li et al 9 studied that, in contrast to normal beliefs, NoCs suf...

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An Edge-Isoperimetric Problem for Powers of the Petersen Graph

Cited by 67 publications

References 13 publications

Embedding of Recursive Circulants into Certain Necklace Graphs

Embedding of Recursive Circulants into Certain Necklace Graphs

Linear Wirelength of Folded Hypercubes

Reconfigurable Fault‐tolerance mapping of ternary N‐cubes onto chips

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