Results and Problems Concerning Edge Transformed Graphs

Let X be a simple graph of order n. By an edge transformed graph we mean a graph Y obtained from X by either the insertion or the deletion of a single edge of X. Studies of this graph operation have been carried out in a variety of contexts. These have been in both probabilistic and deterministic settings. We shall present results and problems concerning properties of graphs whose vertices are unlabeled graphs where two vertices are adjacent if and only if their associated graphs are one-edge transformations of each other. Note that the graph theoretical distance between two vertices in such a graph is the least number of insertions and/or deletions needed to obtain one associated graph from the other associated graph. Also note that the degree of a vertex in such graphs is the number of nonisomorphic one-edge extended and/or deleted graphs that can be obtained from the associated graph Two areas of study concerning these graphs are e determination of the diameter radius center and periphery of such graphs whose vertex sets are various sets of graphs, and e determination of the number of one-edge extensions of a given associated graph and its structure when the number of extensions is maximum among a specified set of graphs.