Complexity Dichotomies for the Minimum $$\mathcal {F}$$-Overlay Problem

For a (possibly infinite) fixed family of graphs \(\mathcal {F}\), we say that a graph G overlays \(\mathcal {F}\) on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of \(\mathcal {F}\) as a spanning subgraph. While it is easy to see that the complete graph on |V(H)| overlays \(\mathcal {F}\) on a hypergraph H whenever the problem admits a solution, the Minimum \(\mathcal {F}\)-Overlay problem asks for such a graph with the minimum number of edges. This problem allows to generalize some natural problems which may arise in practice. For instance, if the family \(\mathcal {F}\) contains all connected graphs, then Minimum \(\mathcal {F}\)-Overlay corresponds to the Minimum Connectivity Inference problem (also known as Subset Interconnection Design problem) introduced for the low-resolution reconstruction of macro-molecular assembly in structural biology, or for the design of networks.