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Embedding cycles and paths on solid grid graphs

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Abstract

Despite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if \(k\le L\) and \(k\equiv L (\mathrm{mod}\ 2)\), in which L is the length of the longest path between s and t. Our presented two algorithms show that such embeddings can be found in linear time for cycles and quadratic time for paths, with respect to the size of graph G. In the case of rectangular grid graphs, the running time of the algorithms can be improved to O(k) and O\((k^2)\), respectively. In addition, we extend our results to \(m\times n\times o\) 3D grids. A application of our result is in the interconnection network mapping in parallel processing.

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Authors and Affiliations

  1. Department of Computer Engineering and IT, Amirkabir University of Technology, Tehran, Iran

    Asghar Asgharian Sardroud & Alireza Bagheri

  2. Department of Computer Engineering, Tehran North Branch, Islamic Azad University, Tehran, Iran

    Alireza Bagheri

Authors
  1. Asghar Asgharian Sardroud
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  2. Alireza Bagheri
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Correspondence to Alireza Bagheri.

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Asgharian Sardroud, A., Bagheri, A. Embedding cycles and paths on solid grid graphs. J Supercomput 73, 1322–1336 (2017). https://doi.org/10.1007/s11227-016-1811-y

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  • DOI: https://doi.org/10.1007/s11227-016-1811-y

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