On embedding a graph in the grid with the minimum number of bends
SIAM Journal on Computing
Counting embeddings of planar graphs using DFS trees
SIAM Journal on Discrete Mathematics
Exact solution of large-scale, asymmetric traveling salesman problems
ACM Transactions on Mathematical Software (TOMS)
SIAM Journal on Computing
An experimental comparison of four graph drawing algorithms
Computational Geometry: Theory and Applications
A new approximation algorithm for the planar augmentation problem
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Optimizing over All Combinatorial Embeddings of a Planar Graph
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
Computing Orthogonal Drawings with the Minimum Number of Bends
WADS '97 Proceedings of the 5th International Workshop on Algorithms and Data Structures
On the Compuational Complexity of Upward and Rectilinear Planarity Testing
GD '94 Proceedings of the DIMACS International Workshop on Graph Drawing
A note on computing a maximal planar subgraph using PQ-trees
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Bend Minimization in Orthogonal Drawings Using Integer Programming
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
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We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. At IPCO' 99 we presented a first characterization of the set of all possible embeddings of a given biconnected planar graph G by a system of linear inequalities. This system of linear inequalities can be constructed recursively using SPQR-trees and a new splitting operation. In general, this approach may not be practical in the presence of high degree vertices. In this paper, we present an improvement of the characterization which allows us to deal efficiently with high degree vertices using a separation procedure. The new characterization exposes the connection with the asymmetric traveling salesman problem thus giving an easy proof that it is NP-hard to optimize arbitrary objective functions over the set of combinatorial embeddings. Computational experiments on a set of over 11000 benchmark graphs show that we are able to solve the problem for graphs with 100 vertices in less than one second and that the necessary data structures for the optimization can be build in less than 12 seconds.