Intersection-Representation by Connected Subgraphs of Some \(n\)-Cyclomatic Graph

A hypergraph \(H\) is called connected over a graph \(G\) with the same vertex set as \(H\) if every hyperedge of \(H\) induces a connected subgraph in \(G\). A graph \(F\) is representable in the graph \(G\) if there is some hypergraph \(H\) which is connected over \(G\) and has \(F\) as its intersection graph. Generalizing the well-known problem of representability in forests, the following problems are investigated: Which hypergraphs are connected over some \(n\)-cyclomatic graph, and which graphs are representable in some \(n\)-cyclomatic graph, for any fixed integer \(n\)? Several notions developed in the theory of subtree hypergraphs and chordal graphs (i.e. in the case \(n = 0\)) yield necessary or sufficient conditions, and in certain special cases even characterizations.