Inserting Multiple Edges into a Planar Graph
| Authors | |
|---|---|
| Year of publication | 2023 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Graph Algorithms and Applications |
| MU Faculty or unit | |
| Citation | |
| web | https://jgaa.info/accepted/2023/631.pdf |
| Doi | http://dx.doi.org/10.7155/jgaa.00631 |
| Keywords | crossing number; multiple edge insertion; fixed parameter tractability |
| Description | Let G be a connected planar (but not yet embedded) graph and F a set of edges with ends in V(G) and not belonging to E(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. A solution to this problem is known to approximate the crossing number of the graph G+F, but unfortunately, finding an exact solution to MEI is NP-hard for general F. The MEI problem is linear-time solvable for the special case of |F|=1 (SODA 01 and Algorithmica), and there is a polynomial-time solvable extension in which all edges of F are incident to a common vertex which is newly introduced into G (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 and JoCO). We present a fixed-parameter algorithm for the MEI problem in the case that G is biconnected, which is extended to also cover the case of connected G with cut vertices of bounded degree. These are the first exact algorithms for the general MEI problem, and they run in time O(|V(G)|) for any constant k. |
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