Crossing Minimization in the Straight-Line Embedding of Graphs.

The problem of embedding a graph in the plane with the minimum number of edgecrossings arises in some circuit layout problems. It has been known that this problem is in general NP-hard. An interesting and relevant problem in the area of book embedding was recently shown to be NP-hard. This result implies that the former problem is NP-hard even when the vertices are placed on a straight line l and the edges are drawn completely on either side of l. In this paper, we show that the problem remains NP-hard even if, in addition to these constraints, the positions of the vertices on SCRIPT l are predetermined