Short Topological Decompositions of Non-Orientable Surfaces

TL;DR

This paper presents a polynomial-time algorithm to compute short, canonical systems of loops in non-orientable surfaces, advancing understanding of their topological decompositions and confirming a special case of Negami's conjecture.

Contribution

It introduces the first polynomial-time algorithm for canonical non-orientable systems of loops with bounded intersections, solving an open problem in non-orientable surface topology.

Findings

Algorithm computes loops intersecting edges at most 30 times

Confirms a special case of Negami's conjecture on joint crossing numbers

Provides a correction to Negami's previous argument

Abstract

In this article, we investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface computes a canonical non-orientable system of loops so that any loop from the canonical system intersects any edge of the graph in at most 30 points. The existence of such short canonical systems of loops was well known in the orientable case and an open problem in the non-orientable case. Our proof techniques combine recent work of Schaefer-\v{S}tefankovi\v{c} with ideas coming from computational biology, specifically from the signed reversal distance algorithm of Hannenhalli-Pevzner. The existence of short canonical non-orientable systems of loops confirms a special case of a conjecture of Negami on the joint crossing number of two embeddable graphs. We also provide a correction for an argument of Negami bounding the joint crossing number of two non-orientable graph embeddings.