# The Degenerate Crossing Number and Higher-Genus Embeddings

## Abstract

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*) \(\le \) 6 gcr(*G*), and dcr(*G*) = gcr(*G*) as long as dcr(*G*) \(\le \) 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 \(\mathrm {\mathbf {NP}}\)-complete.

### Keywords

Degenerate crossing number Non-orientable genus Genus crossing number## Notes

### Acknowledgments

We would like to thank Bojan Mohar for suggesting the question, and giving us detailed feedback on earlier drafts of this paper. We are also grateful for helpful comments by the anonymous reviewers.

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