Loop and cyclic niche graphs

The niche graph of a digraph is a graph created by taking the vertices of the digraph and connecting any pair which occur in an in-neighborhood or an out-neighborhood of the digraph. Originally the problem of which graphs were the niche graphs of acyclic digraphs received much attention; this paper considers the effect of relaxing the requirement that the digraph be acyclic. Several new classes of graphs turn out to be niche graphs under this relaxation, but unlike the analogous result for competition graphs, many graphs still are not niche graphs, nor can they be made into niche graphs by adding isolated vertices.