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? -- Hauke Reddmann <:-EX8 email@example.com Die Weltformel: IQ+dB=const.