Reconstructing graphs as subsumed graphs of hypergraphs, and some self-complementary triple systems

AbstractLetV be a set ofn elements. The set of allk-subsets ofV is denoted $$\left( {_k^V } \right)$$ . Ak-hypergraph G consists of avertex-set V(G) and anedgeset $$E(G) \subseteq \left( {_k^{V(G)} } \right)$$ , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphGv. The graphs {GvνvεV(G)} aresubsumed byG. Each subsumed graphGv is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachGv≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG*, all of whose subsumed graphs are isomorphic toH, whereG andG* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way.