Degenerate crossing number and signed reversal distance

TL;DR

This paper explores the relationship between degenerate crossing number and non-orientable genus in graph embeddings, providing a structure theorem, counterexamples to a conjecture, and applications to genome rearrangement problems.

Contribution

It proves a structure theorem linking degenerate crossing number and non-orientable genus, and applies this to analyze signed permutation reversal distances.

Findings

Counterexample to Mohar's stronger conjecture

Most 2-vertex cases satisfy Mohar's conjecture

Connection established between graph embeddings and genome rearrangement algorithms

Abstract

Given a graph drawn in the plane, the degenerate crossing number of the drawing is the number of points in the plane which are contained in the relative interior of at least two edges, where each edge is required to be drawn as a simple arc. The degenerate crossing number of a graph is the minimum degenerate crossing number among all its drawings. Given a drawing, cutting a neighborhood of the surface around each crossing and pasting a M\"obius band gives a non-orientable surface, on which the drawing of the graph can be extended to an embedding. From this observation, Mohar derived that the degenerate crossing number of a graph is at most its non-orientable genus, and conjectured that these quantities are equal for every graph. He also made a stronger conjecture for loopless pseudo-triangulations with a fixed embedding scheme. In this paper, we prove a structure theorem that allows to understand when the degenerate crossing number and non-orientable genus coincide in a large class of loopless bipartite embedding schemes. In particular, we provide a counterexample to Mohar's stronger conjecture, but show that in the vast majority of the 2-vertex cases, as well as for many bipartite graphs, Mohar's conjecture is satisfied. The reversal distance between two signed permutations is the minimum number of reversals that transform one permutation to the other one. If we represent the trajectory of each element of a signed permutation under successive reversals by a simple arc, we obtain a drawing of a 2-vertex embedding scheme with degenerate crossings. Our main result is proved by leveraging this connection and a classical result in genome rearrangement (the Hannenhalli--Pevzner algorithm) and can also be understood as an extension of this algorithm when the reversals do not necessarily happen in a monotone order.