The transformation graph image when image

The transformation graph image of G is the graph with vertex set image in which the vertex x and y are joined by an edge if one of the following conditions holds: (i) image, and x and y are not adjacent in G, (ii) image, and x and y are adjacent in G, (iii) one of x and y is in image and the other is in image, and they are incident in G. In this paper, it is shown that for two graphs G and image, image if and only if image. Simple necessary and sufficient conditions are given for image to be planar and hamiltonian, respectively. It is also shown that for a graph G, the edge-connectivity of image is equal to its minimum degree. Two related conjectures and some research problems are presented.