Simple graphs occur in several formally distinct guises. This submission establishes precise bridges from $SimpleGraph V$ to five of them: symmetric loopless digraphs, simple loopless multigraphs, spanning 2-uniform hypergraphs, 2-uniform edge set systems, and vertex-indexed neighborhood set systems.

For digraphs, hypergraphs, edge set systems, and neighborhood set systems the comparison is an equivalence of the corresponding types. A simple multigraph is compared with a simple graph up to simultaneous vertex and edge isomorphism, because $Graph α β$ keeps ambient vertex and edge types. Looplessness of the hypergraph and edge-set-system representations is derived from 2-uniformity rather than recorded as redundant data.

Neighborhood systems retain their vertex indexing: the neighborhood assigned to $u$ is exactly the set of vertices adjacent to $u$. This indexing is what makes the construction invertible even when two vertices have the same open neighborhood.

Together these results make the representation choice modular: a statement may be proved in the graph language best suited to it and transported across the appropriate equivalence or isomorphism, subject only to the usual isomorphism-invariance requirement in the multigraph case.