The clique hypergraph of a graph has the same vertices than the graph, and all cliques as hyperedges. Three edges form an asteroidal triple if any two are connected by a path avoiding vertices and neighbors of the third edge. A class H of hypergraphs is closed under subhypergraphs if every hypergraph obtained from a member of H by deleting some hyperedges is again in H . A class G of graphs is closed under subgraphs if every induced subgraph of any member of G is again in G . Dicliques are ordered variants of bicliques.
G have both the same vertex set than G .
Two distinct vertices are adjacent in the square if their distance in G is at most 2, and in the 2-step graph if they have some common neighbor in G .