The niche graph of a digraph \(D\) is the undirected graph defined on the same vertex set in which two vertices are adjacent if they share either a common in-neighbor or a common out-neighbor in \(D\). A hierarchy of graphs exists, depending on the condition of being the niche graph of a digraph having, respectively, no cycles, no cycles of length two, no loops, or loops. Our goal is to classify in this hierarchy all graphs of order \(n \geq 3\) having a generated subgraph isomorphic to the discrete graph on \(n – 2\) vertices.