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 andG′,G−++ G′−++ if and only ifG 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. © 2005 Elsevier B.V. All rights reserved.