Inserting multiple edges into a planar graph

Abstract

Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not in G. The multiple edge insertion problem (MEI) asks for a drawing of G F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem is known to approximate the crossing number of the graph G F . Finding an exact solution to MEI is NP-hard for general F, but linear time solvable for the special case of |F| = 1 (SODA 01, Algorithmica) and polynomial time solvable when all of F are incident to a new vertex (SODA 09). The complexity for general F but with constant k = |F| was open, but algorithms both with relative and absolute approximation guarantees have been presented (SODA 11, ICALP 11). We show that the problem is fixed parameter tractable (FPT) in k for biconnected G, or if the cut vertices of G have bounded degrees. We give the first exact algorithm for this problem; it requires only O(|V (G)|) time for any constant k. © Markus Chimani and Petr Hlinìný.