A polygonal realization has numerical genus zero

For an arbitrary listed finite multigraph with declared incident endpoints, an actual polygonal embedding that literally realizes a clockwise rotation has componentwise numerical genus zero. The proof constructs colors on the actual face quotient from every vertex-even indexed edge label and applies the cycle-space converse.

Coordinates are arbitrary complex numbers and segment parameters range over all real numbers in [0,1]. Loops retain their two ends, parallel edges retain their list indices, and sparse or duplicate declarations, isolates, disconnected graphs and the empty graph are permitted. There is no degree bound or supplied topological, face-color or genus oracle.

This is an original finite PL proof. It does not establish a topological-source comparison, taming of arbitrary topological arcs, Hanani–Tutte, a recognition algorithm or a CSP dichotomy.

Main results

• Cslib.Complexity.Graph.PolygonalDrawing.IsEmbedding.numericalGenusZero.

theorem Cslib.Complexity.Graph.PolygonalDrawing.IsEmbedding.numericalGenusZero {G : Graph} {D : G.PolygonalDrawing} (hD : D.IsEmbedding) {r : G.RotationSystem} (hG : ∀ v ∈ G.incidenceVertices, v ∈ G.vertices) (hR : D.Realizes r) : r.NumericalGenusZero

A genuine finite polygonal embedding with exact literal clockwise realization has numerical genus zero on every component, including isolated declared vertices.