Inserting an Edge into a Planar Graph

Abstract Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e where all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a combinatorial embedding of a planar graph G where the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, that is able to find a solution with the minimum number of crossings.

[1]  Petra Mutzel,et al.  Optimizing over All Combinatorial Embeddings of a Planar Graph , 1999, IPCO.

[2]  Petra Mutzel,et al.  Computing Optimal Embeddings for Planar Graphs , 2000, COCOON.

[3]  Roberto Tamassia,et al.  On Embedding a Graph in the Grid with the Minimum Number of Bends , 1987, SIAM J. Comput..

[4]  Michael Jünger,et al.  A note on computing a maximal planar subgraph using PQ-trees , 1998, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[5]  Farhad Shahrokhi,et al.  Crossing Numbers of Graphs, Lower Bound Techniques , 1994, GD.

[6]  Petra Mutzel,et al.  A Linear Time Implementation of SPQR-Trees , 2000, GD.

[7]  Walter Didimo,et al.  Computing Orthogonal Drawings with the Minimum Number of Bends , 1997, IEEE Trans. Computers.

[8]  Roberto Tamassia,et al.  On the Compuational Complexity of Upward and Rectilinear Planarity Testing , 1994, Graph Drawing.

[9]  Karlheinz Ochs,et al.  Generation of wave digital structures for connection networks containing ideal transformers , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[10]  David Harel,et al.  Randomized Graph Drawing with Heavy-Duty Preprocessing , 1995, J. Vis. Lang. Comput..

[11]  Giuseppe Liotta,et al.  An Experimental Comparison of Four Graph Drawing Algorithms , 1997, Comput. Geom..

[12]  Petra Mutzel,et al.  An Experimental Study of Crossing Minimization Heuristics , 2003, Graph Drawing.

[13]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[14]  Frank Harary,et al.  Graph Theory , 2016 .

[15]  David S. Johnson,et al.  Crossing Number is NP-Complete , 1983 .

[16]  Petra Mutzel,et al.  Graph Embedding with Minimum Depth and Maximum External Face , 2003, GD.

[17]  David Harel,et al.  Randomized graph drawing with heavy-duty preprocessing , 1994, AVI '94.

[18]  Roberto Tamassia,et al.  On-Line Planarity Testing , 1989, SIAM J. Comput..

[19]  Norishige Chiba,et al.  A Linear Algorithm for Embedding Planar Graphs Using PQ-Trees , 1985, J. Comput. Syst. Sci..

[20]  Roberto Tamassia,et al.  On the Computational Complexity of Upward and Rectilinear Planarity Testing , 1994, SIAM J. Comput..