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
SIAM Journal on Computing
An experimental comparison of four graph drawing algorithms
Computational Geometry: Theory and Applications
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
Computing Optimal Embeddings for Planar Graphs
COCOON '00 Proceedings of the 6th Annual International Conference on Computing and Combinatorics
Bend Minimization in Orthogonal Drawings Using Integer Programming
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
A Linear Time Implementation of SPQR-Trees
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
An Experimental Comparison of Orthogonal Compaction Algorithms (Extended Abstract)
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
Embeddings of planar graphs that minimize the number of long-face cycles
Operations Research Letters
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We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. Our objective function prefers certain cycles of G as face cycles in the embedding. The motivation for studying this problem arises in graph drawing, where the chosen embedding has an important influence on the aesthetics of the drawing. We characterize the set of all possible embeddings of a given biconnected planar graph G by means of a system of linear inequalities with {0,1}- variables corresponding to the set of those cycles in G which can appear in a combinatorial embedding. This system of linear inequalities can be constructed recursively using SPQR-trees and a new splitting operation. Our computational results on two benchmark sets of graphs are surprising: The number of variables and constraints seems to grow only linearly with the size of the graphs although the number of embeddings grows exponentially. For all tested graphs (up to 500 vertices) and linear objective functions, the resulting integer linear programs could be generated within 10 minutes and solved within two seconds on a Sun Enterprise 10000 using CPLEX.