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.
| 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. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Total Graph, Planarity, Transformation Graph
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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
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
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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Download@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}
}
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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