SIAM Journal on Computing
Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas
SIAM Journal on Computing
A Linear Time Implementation of SPQR-Trees
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
Inserting a vertex into a planar graph
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Expander flows, geometric embeddings and graph partitioning
Journal of the ACM (JACM)
Approximating the Crossing Number of Apex Graphs
Graph Drawing
On the crossing number of almost planar graphs
GD'06 Proceedings of the 14th international conference on Graph drawing
Approximating the crossing number of toroidal graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Adding one edge to planar graphs makes crossing number hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Approximating the crossing number of graphs embeddable in any orientable surface
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
An algorithm for the graph crossing number problem
Proceedings of the forty-third annual ACM symposium on Theory of computing
On graph crossing number and edge planarization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Vertex insertion approximates the crossing number of apex graphs
European Journal of Combinatorics
Advances in the planarization method: effective multiple edge insertions
GD'11 Proceedings of the 19th international conference on Graph Drawing
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Let G be a planar 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. As an exact solution to MEI is NP-hard for general F, we present the first approximation algorithm for MEI which achieves an additive approximation factor (depending only on the size of F and the maximum degree of G) in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph G + F, while computing the crossing number of G + F exactly is NP-hard already when |F| = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.