Recognizing random intersection graphs Original Research Article

A polynomial-time algorithm is given which succeeds in reconstructing the simple k-uniform hypergraph H from its ℓ-intersection graph, for almost all random k-uniform hypergraphs H=Hk(n,p), where p≻n−1/2+ε, ε>0. Two related algorithms reconstruct almost every random graph G=G(n,p) from its k-line graph Lk(G) (which is the (k−1)-intersection graph of the set of all complete subgraphs on k vertices), and almost every random graph G from its (k−1)-in-k graph Φk−1,k(G) (which has all complete (k−1)-vertex subgraphs of G as vertices, two of them adjacent if they lie in some common complete k-vertex subgraph), for p≻n−1/k+ε, respectively, p≻n−1/(2k−2)+ε, ε>0.