The realizability of discs with ribbons on a M\"obius strip

TL;DR

This paper provides a quadratic-time criterion for determining when a disk with ribbons, derived from hieroglyphs, can be embedded in a M"obius strip, extending Mohar's realizability criterion.

Contribution

It introduces a new criterion for weak realizability of hieroglyph-based disks on M"obius strips, with an efficient quadratic algorithm.

Findings

Provides a quadratic algorithm for weak realizability.

Extends Mohar's criterion to M"obius strip embeddings.

Establishes a necessary and sufficient condition for realizability.

Abstract

An hieroglyph on n letters is a cyclic sequence of the letters 1,2, . . . , n of length 2n such that each letter appears in the sequence twice.Take an hieroglyph H. Take a convex polygon with 2n sides. Put the letters in the sequence of letters of the hieroglyph on the sides of the convexpolygon in the same order. For each letter i glue the ends of a ribbon to thepair of sides corresponding to the letter i. Call the resulting surface a disk with ribbons corresponding to the hieroglyph H. An hieroglyph H is weakly realizable on the M\"obius strip if some disk with ribbons corresponding to H can be cut out of the M\"obius strip. We give a criterion for weak realizability, which gives a quadratic (in the number of letters) algorithm. Our criterion is based on the Mohar criterion for realizability of a disk with ribbons in the M\"obius strip.