Degenerate crossing number and signed reversal distance

The degenerate crossing number of a graph is the minimum number of transverse crossings among all its drawings, where edges are represented as simple arcs and multiple edges passing through the same point are counted as a single crossing. Interpreting each crossing as a cross-cap induces an embedding into a non-orientable surface. In 2007, Mohar showed that the degenerate crossing number of a graph is at most its non-orientable genus and he conjectured that these quantities are equal for every graph. He also made the stronger conjecture that this also holds for any loopless pseudotriangulation with a fixed embedding scheme. In this paper, we prove a structure theorem that almost completely classifies the loopless 2-vertex embedding schemes for which the degenerate crossing number equals the non-orientable genus. In particular, we provide a counterexample to Mohar's stronger conjecture, but show that in the vast majority of the 2-vertex cases, the conjecture does hold. 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 Hannenhali-Pevzner algorithm) and can also be understood as an extension of this algorithm when the reversals do not necessarily happen in a monotone order.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Bundled Crossings Revisited

International audienceAn effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into bundles. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a bundled crossing. We consider the problem of bundled crossing minimization: A graph is given and the goal is to find a bundled drawing with at most k bundled crossings. We show that the problem is NP-hard when we require a simple drawing. Our main result is an FPT algorithm (in k) for simple circular layouts where vertices must be placed on a circle and edges must be drawn inside the circle. These results make use of the connection between bundled crossings and graph genus. We also consider bundling crossings in a given drawing, in particular for storyline visualizations

The Degenerate Crossing Number and Higher-Genus Embeddings

If a graph embeds in a surface with k crosscaps, does it always have an embedding in the same surface in which every edge passes through each crosscap at most once? This well-known open problem can be restated using crossing numbers: the degenerate crossing number, dcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps in which every edge passes through each crosscap at most once. The genus crossing number, gcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps. The question then becomes whether dcr(G) = gcr(G), and it is in this form that it was first asked by Mohar. We show that dcr(G) ≤ 6 gcr(G), and dcr(G) = gcr(G) as long as dcr(G) ≤ 3. We can separate dcr and gcr for (single-vertex) graphs with embedding schemes, but it is not clear whether the separating example can be extended into separations on simple graphs. Finally, we show that if a graph can be embedded in a surface with crosscaps, then it has an embedding in that surface in which every edge passes through each crosscap at most twice. This implies that dcr is NP-complete