Crossing Minimization in Linear Embeddings of Graphs
IEEE Transactions on Computers
Geometric symmetry in graphs
Bounds for rectilinear crossing numbers
Journal of Graph Theory
Spring algorithms and symmetry
Theoretical Computer Science - computing and combinatorics
Symmetric drawings of triconnected planar graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Which Aesthetic has the Greatest Effect on Human Understanding?
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
Detecting Symmetries by Branch & Cut
GD '01 Revised Papers from the 9th International Symposium on Graph Drawing
The optimal linear arrangement problem: algorithms and approximation
The optimal linear arrangement problem: algorithms and approximation
Geometric automorphism groups of graphs
Discrete Applied Mathematics
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We consider the problem of drawing a graph with a given symmetry such that the number of edge crossings is minimal. We show that this problem is NP-hard, even if the order of orbits around the rotation center or along the reflection axis is fixed. Nevertheless, there is a linear time algorithm to test planarity and to construct a planar embedding if possible. Finally, we devise an O (m log m) algorithm for computing a crossing minimal drawing if inter-orbit edges may not cross orbits, showing in particular that intra-orbit edges do not contribute to the NP-hardness of the crossing minimization problem for symmetries. From this result, we can derive an O (m log m) crossing minimization algorithm for symmetries with an orbit graph that is a path.