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The degenerate crossing number of a graph is the minimum number of transversecrossings among all its drawings, where edges are represented as simple arcsand multiple edges passing through the same point are counted as a singlecrossing. Interpreting each crossing as a cross-cap induces an embedding into anon-orientable surface. In 2007, Mohar showed that the degenerate crossingnumber of a graph is at most its non-orientable genus and he conjectured thatthese quantities are equal for every graph. He also made the strongerconjecture that this also holds for any loopless pseudotriangulation with afixed embedding scheme.  In this paper, we prove a structure theorem that almost completely classifiesthe loopless 2-vertex embedding schemes for which the degenerate crossingnumber equals the non-orientable genus. In particular, we provide acounterexample to Mohar's stronger conjecture, but show that in the vastmajority of the 2-vertex cases, the conjecture does hold.  The reversal distance between two signed permutations is the minimum numberof reversals that transform one permutation to the other one. If we representthe trajectory of each element of a signed permutation under successivereversals by a simple arc, we obtain a drawing of a 2-vertex embedding schemewith degenerate crossings. Our main result is proved by leveraging thisconnection and a classical result in genome rearrangement (theHannenhali-Pevzner algorithm) and can also be understood as an extension ofthis algorithm when the reversals do not necessarily happen in a monotoneorder.

Degenerate crossing number and signed reversal distance