Homology Betti numbers of intersection graphs Under certain conditions, the topology of the union of the hyperedges is to some extend reflected in the structure of its intersection graph. This is the reason why the octahedron is no intersection graph of rectangles in the plane, but of unit disks, for instance. We compare the topology of this complex with the topology of the union of all sets in the geometric case. This follows by classical topological theorems.
Examples are straight lines, or unit-disks, or boxes.
Nerve and union are homological equivalent for every family of subsets of R d obeying the Leray-property. The second follows immediately since higher homology groups of subsets of R d vanish. An example is the class of intersection graphs of boxes. See here for examples and problems. Discrete models For discrete models, we only need the Helly-property .