Abstract

Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest non-contractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest non-null-homologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest non-separating cycle.

Citations

239	Handbook of Graph Theory - Gross, Yellen - 2004
161	Graphs on Surfaces - Mohar, Thomassen - 2001
137	Faster shortestpath algorithms for planar graphs - Henzinger, Klein, et al. - 1997
133	Classical Topology and Combinatorial Group Theory - Stillwell - 1980
106	Fast algorithms for shortest paths in planar graphs, with applications - Frederickson - 1987
88	Topological noise removal - Guskov, Wood
60	Removing excess topology from isosurfaces - Wood, Hoppe, et al.
37	Embeddings of graphs with no short noncontractible cycles - Thomassen - 1990
36	Multiple-source shortest paths in planar graphs - Klein - 2005
33	V.V.: Matching is as easy as matrix inversion. Combinatorica 7 - Mulmuley, Vazirani, et al. - 1987
32	Finding shortest non-separating and non-contractible cycles for topologically embedded graphs - Cabello, Mohar - 2005
28	Dynamic generators of topologically embedded graphs - Eppstein
23	Multiple source shortest paths in a genus g graph - Cabello, Chambers - 2007
22	Algebraic Topology. Cambridge Univ - Hatcher - 2002
22	Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time - Kutz - 2006
22	Minimum s-t cut of a planar undirected network in O(n log 2 n) time - Reif - 1983
20	Approximation algorithms via contraction decomposition - Demaine, Hajiaghayi, et al. - 2007
19	Splitting (complicated) surfaces is hard - Chambers, Verdière, et al. - 2006
18	Optimally cutting a surface into a disk. Discrete Comput - Erickson, Har-Peled
17	Computing crossing number in linear time - Kawarabayashi, Reed - 2007
17	Maximum (s, t)-flows in planar networks in o(|v|log|v|) time - Weihe - 1997
16	Maximum flow in planar networks - Itai, Shiloach - 1979
15	An O(n log n) algorithm for maximum stflow in a directed planar graph - Borradaile, Klein - 2009
14	Minimum cuts and shortest homologous cycles - Erickson, Chambers, et al. - 2009
12	Graph and map isomorphism and all polyhedral embeddings in linear time - KAWARABAYASHI, MOHAR
11	Homology flows, cohomology cuts - Erickson, Chambers, et al. - 2009
10	Finding one tight cycle - Cabello, DeVos, et al.
9	Probabilistic embeddings of bounded genus graphs into planar graphs - Indyk, Sidiropoulos
8	Polynomial-time approximation schemes for subset-connectivity problems in bounded genus graphs - Borradaile, Demaine, et al. - 2009
7	Computing the shortest essential cycle - Erickson, Worah - 2010
7	Minimum cut in directed planar networks - Janiga, Koubek - 1992
6	Quantifying homology classes II: Localization and stability - Chen, Freedman - 2007
6	Minimum cuts and shortest non-separating cycles via homology covers - Erickson, Nayyeri - 2011
5	Randomly removing g handles at once - Borradaile, Lee, et al. - 2009
5	Multiple-source shortest paths in embedded graphs, Preprint, February 2012, ArXiv:1202.0314. Full version of [26 - Cabello, Chambers, et al.
5	Finding shortest non-trivial cycles in directed graphs on surfaces - Cabello, Verdière, et al. - 2010
5	Outputsensitive algorithm for the edge-width of an embedded graph - Cabello, Verdière, et al. - 2010
5	Maximum flows and parametric shortest paths in planar graphs - Erickson - 2010
5	Improved algorithms for min cut and max flow in undirected planar graphs - Italiano, Nussbaum, et al.
3	Finding shortest contractible and shortest separating cycles in embedded graphs - Cabello
3	Shortest non-trivial cycles in directed surface graphs - Erickson
3	Minimum s − t cut in undirected planar graphs when the source and the sink are close - Kaplan, Nussbaum
2	Finding cycles with topological properties in embedded graphs - Cabello, Verdière, et al. - 2010
1	Global minimum cuts in surface embedded graphs - Erickson, Fox, et al. - 2012