Planarizing an Unknown Surface

It has been recently shown that any graph of genus g > 0 can be stochastically embedded into a distribution over planar graphs, with distortion O(log(g + 1)) [Sidiropoulos, FOCS 2010]. This embedding can be computed in polynomial time, provided that a drawing of the input graph into a genus-g surface is given.

[1]  J. M. Sek.  On embedding trees into uniformly convex Banach spaces , 1999 .

[2]  Glencora Borradaile,et al.  Randomly removing g handles at once , 2010, Comput. Geom..

[3]  Carsten Thomassen,et al.  The Graph Genus Problem is NP-Complete , 1989, J. Algorithms.

[4]  James R. Lee,et al.  Improved approximation algorithms for minimum-weight vertex separators , 2005, STOC '05.

[5]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[6]  Jianer Chen,et al.  A Note on Approximating Graph Genus , 1997, Inf. Process. Lett..

[7]  Erin W. Chambers,et al.  Homology flows, cohomology cuts , 2009, STOC '09.

[8]  Anastasios Sidiropoulos,et al.  Optimal Stochastic Planarization , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[9]  Piotr Indyk,et al.  Probabilistic embeddings of bounded genus graphs into planar graphs , 2007, SCG '07.

[10]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[11]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[12]  Bruce A. Reed,et al.  A Simpler Linear Time Algorithm for Embedding Graphs into an Arbitrary Surface and the Genus of Graphs of Bounded Tree-Width , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[13]  David Eppstein,et al.  Dynamic generators of topologically embedded graphs , 2002, SODA '03.

[14]  Carsten Thomassen.  Triangulating a Surface with a Prescribed Graph , 1993, J. Comb. Theory, Ser. B.

[15]  Amit Sahai,et al.  Dimension reduction in the /spl lscr//sub 1/ norm , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[16]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[17]  Mikkel Thorup.  Compact oracles for reachability and approximate distances in planar digraphs , 2004, JACM.

[18]  Jeff Erickson,et al.  Greedy optimal homotopy and homology generators , 2005, SODA '05.

[19]  A. Sahai,et al.  Dimension Reduction in the l 1 norm , 2002 .

[20]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[21]  Piotr Indyk,et al.  Algorithmic applications of low-distortion geometric embeddings , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.