Degrees in a digraph whose nodes are graphs

Abstract By an f - graph we mean an unlabeled graph having no vertex of degree greater than f . Let D ( n , f ) denote the digraph whose node set is the set of f -graphs of order n and such that there is an arc from the node corresponding to graph H to the node corresponding to the graph K if and only if K is obtainable from H by the addition of a single edge. In earlier work, algorithms were developed which produce exact results about the structure of D ( n , f ), nevertheless many open problems remain. For example, the computation of the order and size of D ( n , f ) for a number of values of n and f have been obtained. Formulas for the order and size for f = 2 have also been derived. However, no closed form formulas have been determined for the order and size of D ( n , f ) for any value of f . Here we focus on questions concerning the degrees of the nodes in D ( n , n − 1) and comment on related questions for D ( n , f ) for 2 ⩽ f n − 1.