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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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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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.
Total Graph, Planarity, Transformation Graph
To cite this article
On the Planarity of G++−, American Journal of Applied Mathematics.
Vol. 6, No. 1,
2018, pp. 23-27.
Copyright © 2018 Authors retain the copyright of this article.
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M. Behzad, A criterion for the planarity of the total graph of a graph, Proc. Cambridge Philos. Soc. 63 (1967) 679-681.
J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976.
J. Chen, L. Huang, J. Zhou, Super connectivity and super edge-connectivity of transformation graphs G+ − +, Ars Combin. 105 (2012) 103-115.
A. Deng, A. Kelmans, Laplacian spectra of digraph transformations, Linear Multilinear Algebra 65 (2017) 699–730.
A. Deng, M. Feng, A. Kelmans, Adjacency polynomials of digraph transformations, Discrete Appl. Math. 206 (2016) 15–38.
A. Deng, A. Kelmans, J. Meng, Laplacian spectra of regular graph transformations, Discrete Appl. Math. 161 (2013) 118–133.
J. Li, J. Liu, Some basic properties of a class of total transformation digraphs, Ars Combin. 116 (2014) 205-211.
X. Liu, On the planarity of G− − −, J. Xinjiang Univ. Sci. Eng. 23(2) (2006) 159-161.
B. Wu, J. Meng, Basic properties of total transformation graphs, J. Math. Study 34(2) (2001) 109-116.
B. Wu, L. Zhang, Z. Zhang, The transformation graph Gxyz when xyz=－++, Discrete Math. 296 (2005) 263-270.
L. Xu, B. Wu, Transformation graph G− + −, Discrete Math. 308 (2008) 5144–5148.
L. Yi, B. Wu, The transformation graph G+ + −, Australas. J. Combin. 44 (2009) 37-42.
L. Zhen, B. Wu, Hamiltonicity of transformation graph G+ − −, Ars Combin. 108 (2013) 117-127.