Cyclic Niche Graphs and Grids

A graph \(G = (V, E)\) is a loop niche graph if there is a digraph \(D = (V, A)\)such that \(xy \in E\) iff there exists \(z \in V\) such that either \(xz\) and \(yz \in A\) or \(zx\) and \(zy \in A\). If \(D\) has no loops, \(G\) is a cyclic niche graph, and if \(D\) is acyclic, \(G\) is a niche graph. We give a characterization of triangle-free cyclic niche graphs, and apply this to classify grids.