Numerical genus zero is equivalent to polygonal realization

For a supplied rotation on a listed finite multigraph with declared incident endpoints, the existing componentwise numerical genus-zero predicate is equivalent to an actual polygonal embedding realizing its literal clockwise order. Coordinates are arbitrary complex numbers and contacts quantify over all real segment parameters. Loops, indexed parallel occurrences, arbitrary degree, disconnected components, repeated declarations, isolates and empty graphs are included.

This finite PL equivalence does not prove topological taming, a source-model comparison, Hanani–Tutte, recognition, coordinate-bit or runtime bounds, or a planar CSP dichotomy.

Main results

• Cslib.Complexity.Graph.RotationSystem.numericalGenusZero_iff_exists_polygonalDrawing.

theorem Cslib.Complexity.Graph.RotationSystem.numericalGenusZero_iff_exists_polygonalDrawing {G : Graph} (r : G.RotationSystem) (hG : ∀ v ∈ G.incidenceVertices, v ∈ G.vertices) : r.NumericalGenusZero ↔ ∃ (D : G.PolygonalDrawing), D.IsEmbedding ∧ D.Realizes r

A supplied rotation has componentwise numerical genus zero exactly when an actual polygonal embedding realizes its literal clockwise order.