On the Planarity of G++−
American Journal of Applied Mathematics
Volume 6, Issue 1, February 2018, Pages: 23-27
Received: Feb. 10, 2018; Accepted: Mar. 1, 2018; Published: Mar. 22, 2018
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Authors
Lili Yuan, College of Mathematics and System Sciences, Xinjiang University, Urumqi, P. R. China
Xiaoping Liu, Department of Mathematics, Xinjiang Institute of Engineering, Urumqi, P. R. China
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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.
Keywords
Total Graph, Planarity, Transformation Graph
To cite this article
Lili Yuan, Xiaoping Liu, On the Planarity of G++−, American Journal of Applied Mathematics. Vol. 6, No. 1, 2018, pp. 23-27. doi: 10.11648/j.ajam.20180601.15
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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