Degenerate crossing number and signed reversal distance

TL;DR

The paper investigates the relationship between the degenerate crossing number $cr_{\mathrm{deg}}$ and the non-orientable genus of graphs, framing the problem via cross-cap drawings. It develops a structure theory for 2-vertex loopless embedding schemes, proving that most such schemes satisfy Mohar's conjecture while providing a counterexample to a stronger version. By linking to signed reversal distance and implementing the HP algorithm, the authors extend reversal-distance ideas to monotone cross-cap drawings and derive constructive, polynomial-time methods for producing perfect cross-cap drawings in broad cases. The work also introduces techniques such as the blow-up of cross-caps and a reduction from bipartite to 2-vertex graphs, yielding practical algorithms for obtaining perfect drawings in many graph classes.

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 Möbius 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.

Figures (21)

• Figure 1: A degenerate crossing and a cross-cap placed at this crossing.
• Figure 2: Left: A loopless 2-vertex scheme made of a positive block, in red, consisting of only positive edges, and a negative block, in blue, consisting of negative edges. Middle: a cross-cap drawing showing that it has non-orientable genus $5$. The bold red edge enters a cross-cap twice. Right: A cross-cap drawing where each edge enters each cross-cap at most once requires $6$ cross-caps.
• Figure 3: A perfect cross-cap drawing of Figure \ref{['block example']} with two additional edges.
• Figure 4: A decomposition of a loopless $2$-vertex embedded graph into its (cleaned) portions. This example is used as a running example and will reappear in Figures \ref{['portion tree']} and \ref{['fig:corners']}.
• Figure 5: The only two obstructions to perfect cross-cap drawings. On the left, the two cyclic permutations are $(\bar{2},1)$ and $(1,\bar{2})$, and on the right, they are $(\bar{4},\bar{3},1,2)$ and $(1,2,\bar{3},\bar{4})$, where we use cycle notations and the $\bar{\cdot}$ denotes negative signature.

Theorems & Definitions (58)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Definition 2.1: Embedding scheme
• Lemma 2.1
• proof
• Lemma 2.1
• proof