The transformation graph Gxyz when xyz=-++

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 one of the following conditions holds: (i) x,y∈V(G), and x and y are not 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 incident in G. In this paper, it is shown that for two graphs G and G′, G-++≅G′-++ if and only if G≅G′. Simple necessary and sufficient conditions are given for G-++ to be planar and hamiltonian, respectively. It is also shown that for a graph G, the edge-connectivity of G-++ is equal to its minimum degree. Two related conjectures and some research problems are presented.