In article <ap4910Fk250U1@mid.dfncis.de>, <firstname.lastname@example.org> wrote:
> Warning, incoming lousy ASCII, change to fixed font :-) > > o o o > | | | > | O | > \ /|\ / > X | X > / \|/ \ > | O | > | | | > o o o > > The lines |/\ are tied to the unmovable nodes Oo. > (As you see, four lines come out of O and one out of o.) > X denotes a crossing, which is like a virtual crossing > from knot theory, i.e. you can move it ad lib and any > line over any other. Should they cross in the process, > well duh, then you have more crossings.) > > Can you move the lines around such that no horizontal > line going through this graph cuts more than three > of these lines? I think no, but can you lend me a > formal proof?
Can you move a crossing past an o? If so, the thing is a planar graph: just move the four corner o's, top to bottom and bottom to top.
o o o \|/ O | O /|\ o o o
That fulfils your condition about horizontal lines. But is there some other restriction on moving the crossings?