image saturated graphs

The first part of this paper deals with the properties of image-saturated graphs. It will be shown that for any image-saturated graph, G, image, where image is the graph with vertex set image, with which two vertices are adjacent iff the distance between them, in G, is k. In addition to this, a full description of the set of planar image-saturated graphs, image, will be given. It will be shown that there are only three kinds of such graphs. In the second part of the paper a useful characterization of graphs which are image-saturated and image-free will be given in terms of the adjacency and incidence matrices A and B.