On the Planarity of G

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: C 3 , C 3 + K 1 , P 4 , P 4 + K 1 , P 3 + K 2 , P 3 + K 2 + K 1 , K 1,3 , K 1,3 + K 1 , 3 K 2 , 3 K 2 + K 1 , 3 K 2 + 2 K 1 , C 4 , C 4 + K 1 .