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 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 NPhard 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 $k$.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 DOI:
 10.48550/arXiv.1509.07952
 arXiv:
 arXiv:1509.07952
 Bibcode:
 2015arXiv150907952C
 Keywords:

 Computer Science  Data Structures and Algorithms