Abstract
This paper presents a connection between the problem of drawing a graph with minimum number of edge crossings, and the theory of arrangements of pseudolines. In particular, we show that any given arrangement can be forced to occur in every minimum-crossing drawing of an appropriate graph. Using recent results of Goodman, Pollack and Sturmfels, this yields that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, with minimum number of crossings from among all such drawings. We also study the problem of drawing a graph with polygonal edges. Here we obtain a tight bound on the smallest number of breakpoints which are required in the polygonal lines, in order to achieve the (unrestricted) minimum number of crossings.
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