two-graph is a regular two-graph since each pair of distinct vertices appears together in exactly two triples.

Given a simple graph G = (V,E), the set of triples of the vertex set V whose induced subgraph has an odd number of edges forms a two-graph on the set V. Every two-graph can be represented in this way. This example is referred to as the standard construction of a two-graph from a simple graph.

As a more complex example, let T be a tree with edge set E. The set of all triples of E that are not contained in a path of T form

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