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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
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
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
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