Inserting Multiple Edges into a Planar Graph

TL;DR

This paper studies the multiple edge insertion problem in planar graphs, providing the first fixed parameter tractable and exact algorithms for cases with a small number of edges, improving computational efficiency.

Contribution

It introduces the first exact algorithm for MEI with fixed parameter tractability in certain graph classes, reducing complexity for small edge sets.

Findings

The problem is fixed parameter tractable in the number of edges for biconnected graphs.

An exact algorithm runs in linear time for fixed small number of edges.

The study extends understanding of crossing minimization in planar graph augmentations.

Abstract

Let $G$ be a connected planar (but not yet embedded) graph and $F$ a set of additional edges not yet 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 approximates 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 $|F|=1$ (SODA01, Algorithmica) or when all of $F$ are incident to a new vertex (SODA09). The complexity for general $F$ but with constant $k=|F|$ was open, but algorithms both with relative and absolute approximation guarantees have been presented (SODA11, ICALP11). We show that the problem is fixed parameter tractable (FPT) in $k$ for biconnected $G$, or if the cut vertices of $G$ have degrees bounded by a constant. We give the first exact algorithm for this problem; it requires only $O(|V(G)|)$ time for any constant $k$.