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On the Planarity of G++−

Received: 10 February 2018     Accepted: 1 March 2018     Published: 22 March 2018
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Abstract

Let G be a simple graph. The transformation graph G++− of G is the graph with vertex set V (G) ∪ E (G) in which the vertex x and y are joined by an edge if and only if the following condition holds:(i) x, y ∈ V (G) and x and y are adjacent in G, (ii) x, yE (G), and x and y are adjacent in G, (iii) one of x and y is in V (G) and the other is in E (G), and they are not incident in G. In this paper, it is shown G++− is planar if and only if |E (G)| ≤ 2 or G is isomorphic to one of the following graphs: C3, C3 + K1, P4, P4 + K1, P3 + K2, P3 + K2 + K1, K1,3, K1,3 +K1, 3K2, 3K2 + K1, 3K2 + 2K1, C4, C4 + K1.

Published in American Journal of Applied Mathematics (Volume 6, Issue 1)
DOI 10.11648/j.ajam.20180601.15
Page(s) 23-27
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Total Graph, Planarity, Transformation Graph

References
[1] M. Behzad, A criterion for the planarity of the total graph of a graph, Proc. Cambridge Philos. Soc. 63 (1967) 679-681.
[2] J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976.
[3] J. Chen, L. Huang, J. Zhou, Super connectivity and super edge-connectivity of transformation graphs G+ − +, Ars Combin. 105 (2012) 103-115.
[4] A. Deng, A. Kelmans, Laplacian spectra of digraph transformations, Linear Multilinear Algebra 65 (2017) 699–730.
[5] A. Deng, M. Feng, A. Kelmans, Adjacency polynomials of digraph transformations, Discrete Appl. Math. 206 (2016) 15–38.
[6] A. Deng, A. Kelmans, J. Meng, Laplacian spectra of regular graph transformations, Discrete Appl. Math. 161 (2013) 118–133.
[7] J. Li, J. Liu, Some basic properties of a class of total transformation digraphs, Ars Combin. 116 (2014) 205-211.
[8] X. Liu, On the planarity of G− − −, J. Xinjiang Univ. Sci. Eng. 23(2) (2006) 159-161.
[9] B. Wu, J. Meng, Basic properties of total transformation graphs, J. Math. Study 34(2) (2001) 109-116.
[10] B. Wu, L. Zhang, Z. Zhang, The transformation graph Gxyz when xyz=-++, Discrete Math. 296 (2005) 263-270.
[11] L. Xu, B. Wu, Transformation graph G− + −, Discrete Math. 308 (2008) 5144–5148.
[12] L. Yi, B. Wu, The transformation graph G+ + −, Australas. J. Combin. 44 (2009) 37-42.
[13] L. Zhen, B. Wu, Hamiltonicity of transformation graph G+ − −, Ars Combin. 108 (2013) 117-127.
Cite This Article
  • APA Style

    Lili Yuan, Xiaoping Liu. (2018). On the Planarity of G++−. American Journal of Applied Mathematics, 6(1), 23-27. https://doi.org/10.11648/j.ajam.20180601.15

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    ACS Style

    Lili Yuan; Xiaoping Liu. On the Planarity of G++−. Am. J. Appl. Math. 2018, 6(1), 23-27. doi: 10.11648/j.ajam.20180601.15

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    AMA Style

    Lili Yuan, Xiaoping Liu. On the Planarity of G++−. Am J Appl Math. 2018;6(1):23-27. doi: 10.11648/j.ajam.20180601.15

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  • @article{10.11648/j.ajam.20180601.15,
      author = {Lili Yuan and Xiaoping Liu},
      title = {On the Planarity of G++−},
      journal = {American Journal of Applied Mathematics},
      volume = {6},
      number = {1},
      pages = {23-27},
      doi = {10.11648/j.ajam.20180601.15},
      url = {https://doi.org/10.11648/j.ajam.20180601.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180601.15},
      abstract = {Let G be a simple graph. The transformation graph G++− of G is the graph with vertex set V (G) ∪ E (G) in which the vertex x and y are joined by an edge if and only if the following condition holds:(i) x, y ∈ V (G) and x and y are adjacent in G, (ii) x, y ∈ E (G), and x and y are adjacent in G, (iii) one of x and y is in V (G) and the other is in E (G), and they are not incident in G. In this paper, it is shown G++− is planar if and only if |E (G)| ≤ 2 or G is isomorphic to one of the following graphs: C3, C3 + K1, P4, P4 + K1, P3 + K2, P3 + K2 + K1, K1,3, K1,3 +K1, 3K2, 3K2 + K1, 3K2 + 2K1, C4, C4 + K1.},
     year = {2018}
    }
    

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  • TY  - JOUR
    T1  - On the Planarity of G++−
    AU  - Lili Yuan
    AU  - Xiaoping Liu
    Y1  - 2018/03/22
    PY  - 2018
    N1  - https://doi.org/10.11648/j.ajam.20180601.15
    DO  - 10.11648/j.ajam.20180601.15
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 23
    EP  - 27
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20180601.15
    AB  - Let G be a simple graph. The transformation graph G++− of G is the graph with vertex set V (G) ∪ E (G) in which the vertex x and y are joined by an edge if and only if the following condition holds:(i) x, y ∈ V (G) and x and y are adjacent in G, (ii) x, y ∈ E (G), and x and y are adjacent in G, (iii) one of x and y is in V (G) and the other is in E (G), and they are not incident in G. In this paper, it is shown G++− is planar if and only if |E (G)| ≤ 2 or G is isomorphic to one of the following graphs: C3, C3 + K1, P4, P4 + K1, P3 + K2, P3 + K2 + K1, K1,3, K1,3 +K1, 3K2, 3K2 + K1, 3K2 + 2K1, C4, C4 + K1.
    VL  - 6
    IS  - 1
    ER  - 

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Author Information
  • College of Mathematics and System Sciences, Xinjiang University, Urumqi, P. R. China

  • Department of Mathematics, Xinjiang Institute of Engineering, Urumqi, P. R. China

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