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Abstract

We investigate the {\em constrained crossing minimization problem} for graphs defined as follows. Given a connected, planar graph $G=(V,E)$, a combinatorial embedding $\Pi(G)$ of $G$, and a set of pairwise distinct edges $F\subseteq V\times V$, find a drawing of $G\cup F$ in the plane such that the combinatorial embedding $\Pi(G)$ of $G$ is preserved and the number of edge crossings is minimum. This problem arises in the context of automatic graph drawing. Here, the so--called planarization method transforms a general graph into a planar graph and then applies planar graph drawing methods to it. First we show NP--hardness of this problem. Then we formulate an $|F|$--pairs shortest walks problem on an extended dual graph, where the number of crossings between the walks is added to the cost function. We show that this dual problem is equivalent to our original problem. Our approach to solve the dual problem is based on polyhedral combinatorics. We investigate an ILP--formulation and present first computational results using a branch--and--cut algorithm based on ABACUS.