Recognizing a totally odd K4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements

Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Volumn , Issue , 2010, Pages 318-328

  Kawarabayashi, Ken Ichi ^a   Li, Zhentao ^b   Reed, Bruce ^b,c  

^a NATIONAL INSTITUTE OF INFORMATICS   (Japan)

^b MCGILL UNIVERSITY   (Canada)

^c UNIVERSITY OF NICE SOPHIA ANTIPOLIS   (France)

Author keywords

Divide and conquer; Parity disjoint paths; Totally odd K4 subdivision

Indexed keywords

3-CONNECTED GRAPH; ACKERMANN FUNCTIONS; DISJOINT PATHS; DIVIDE AND CONQUER; INPUT EDGES; INPUT GRAPHS; ODD LENGTH; POLYNOMIAL ALGORITHM; POLYNOMIAL-TIME; POLYNOMIAL-TIME ALGORITHMS; RECOGNITION ALGORITHM; VERTEX DISJOINT PATHS;

COMBINATORIAL OPTIMIZATION; POLYNOMIAL APPROXIMATION; PROBLEM SOLVING;

ALGORITHMS;

EID: 77951698501     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (63)

1. S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math. 23 (1989), 11-24.
2. T. Böhme, K. Kawarabayashi, J. Maharry and B. Mohar, Linear connectivity forces large complete bipartite minors, J. Combin. Theory Ser. B. 99 (2009), 557-582.
3. H. L. Bodlaender, A linear-time algorithm for finding tree-decomposition of small treewidth, SIAM J. Comput. 25 (1996), 1305-1317.
4. H. L. Bodlaender, Dynamic Programming Algorithms on Graphs of Bounded Tree-Width, In Proceedings of the 15th International Colloquium on Automata, Languages and Programming, volume 317 (1988), 105-118.
5. V. Chvátal, On certain polytopes associated with graphs, J. Combin. Theory Ser. B 18 (1975), 138-154.
6. M. Chudnovsky, J. Geelen, B. Gerards, L. Goddyn, M. Lohman and P. Seymour, Packing non-zero A-paths in group labelled graphs, Combinaorica 26 (2006), 521-532.
7. B. Courcelle and J. A. Makowsky and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory of Computing Systems 33 (2000), 125-150.
8. I. Dejter and V. Neumann-lara, Unboundedness for generalized odd cycle traversability and a Gallai conjecture, paper presented at the Fourth Caribbean Conference on Computing, Puerto Rico, 1985.
9. E. D. Demaine, M. Hajiaghayi, and K. Kawarabayashi, Algorithmic graph minor theory: Decomposition, approximation and coloring, Proc. 46th Ann. IEEE Symp. Found. Comp. Sci., Pittsburgh, PA, 2005, pp. 637-646.
10. R. Diestel, K. Yu. Gorbunov, T. R. Jensen, and C. Thomassen, Highly connected sets and the excluded grid theorem, J. Combin. Theory Ser. B 75 (1999), 61-73.
11. G. A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc., 27 (1952), 85-92.
12. P. Erdös and L. Pósa, On the maximal number of disjoint circuits of a graph, Publ. Math. Debrecen 9 (1962), 3-12.
13. J. Geelen, B. Gerards, B. Reed, P. Seymour and A. Vetta, On the odd variant of Hadwiger's conjecture, J. Combin. Theory Ser. B 99 (2009), 20-29
14. 4. J. of Combin. Theory, Ser. B 47 (1989), 330-348.
15. R. Halin, S-function for graphs, J. Geometry 8 (1976), 171-186.
16. H. Hadwiger, Über eine Klassifikation der Streckenkomplexe, Vierteljahrsschr. Naturforsch. Ges. Zürich, 88 (1943), 133-142.
17. T. Ibaraki and H. Nagamochi, A linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph, Algorithmica 7, (1992), 583-596.
18. T. R. Jensen and F.B. Shepherd, Note on a conjecture of Toft, Combinatorica 15 (1995), 373-377.
19. R. Kapadia, K. Kawarabayashi, Y. Kobayashi, Z. Li and B. Reed, Faster algorithms for the disjoint paths problem submitted.
20. K. Kawarabayashi, On the connectivity of minimal counterexamples to Hadwiger's conjecture, J. Combin. Theory Ser. B, 97 (2007), 144-150.
21. K. Kawarabayashi, Rooted minors problem in highly connected graphs, Discrete Math 287 (2004) 121-123.
22. K. Kawarabayashi, Improved alogirthm for find a cycle through elements, IPCO'08, 374-384.
23. k-minors, J. Combin. Theory Ser. B. 99 (2009), 738-741.
24. K. Kawarabayashi and B. Reed, Highly parity linked graphs, Combinatorica 29 (2009), 215-225.
25. K. Kawarabayashi and Z. Song, Some remarks on the odd Hadwiger's conjecture, Combinatorica. 27 (2007), 429-438.
26. K. Kawarabayashi and P. Wollan, Non-zero disjoint cycles in highly connected group-labelled graphs, J. Combin. Theory Ser. B. 96, (2006), 296-301.
27. K. Kawarabayashi and A. Nakamoto, The Erdocombining double acute accents-Pósa property for odd cycles on an orientable fixed surface, Discrete Math. 307 (2007), 764-768.
28. K. Kawarabayashi and B. Reed, A nearly linear time algorithm for the half disjoint paths packing, ACM-SIAM Symposium on Discrete Algorithms (SODA'08), 446-454.
29. K. Kawarabayashi, Y. Kobayashi and B. Reed, The disjoint paths problem in quaratic time, submitted.
30. 4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements (journal version) in preparation (draft available at url)
31. R. M. Karp, On the computational complexity of combinatorial problems, Network, 5, (1975), 45-68.
32. J. Kleinberg, Decision algorithms for unsplittable flows and the half-disjoint paths problem, Proc. 30th ACM Symposium on Theory of Computing, 1998. 530-539.
33. A. Kostochka, Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica 4 (1984), 307-316.
34. A. Kostochka, The minimum Hadwiger number for graphs with a given mean degree of vertices (in Russian), Metody Diskret Analiz. 38 (1982), 37-58.
35. M. Middendorf and F. Pfeiffer, On the complexity of the disjoint paths problem, Combinatorica, 13 (1993), 97-107.
36. B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, Baltimore, MD, 2001.
37. L. Perkovic and B. Reed, An improved algorithm for finding tree decompositions of small width; International Journal on the Foundations of Computing Science., 11, (2000), 81-85.
38. B. Reed, Finding approximate separators and computing tree width quickly; STOC 1992, Victoria B.C., 1992.
39. B. Reed, Tree width and tangles: a new connectivity measure and some applications, in "Surveys in Combinatorics, 1997 (London)", London Math. Soc. Lecture Note Ser. 241, Cambridge Univ. Press, Cambridge, 1997, pp. 87-162.
40. B. Reed, Mangoes and blueberries, Combinatorica 19 (1999), 267-296.
41. B. Reed, Rooted Routing via Graph Minors, to appear.
42. N. Robertson and P. D. Seymour, Graph minors. II. Algorithmic aspects of tree-width, J. Algorithm 7 (1986), 309-322.
43. N. Robertson and P. D. Seymour, Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), 92-114.
44. N. Robertson and P. D. Seymour, Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B 63 (1995), 65-110.
45. N. Robertson and P. D. Seymour, Graph minors. XVI. Excluding a non-planar graph, J. Combin. Theory Ser. B 89 (2003), 43-76.
46. N. Robertson and P. D. Seymour, An outline of a disjoint paths algorithm, in: "Paths, Flows, and VLSI-Layout," B. Korte, L. Lovász, H. J. Prömel, and A. Schrijver (Eds.), Springer-Verlag, Berlin, 1990, pp. 267-292.
47. N. Robertson, P. D. Seymour and R. Thomas, Quickly excluding a planar graph, J. Combin. Theory Ser. B 62 (1994), 323-348.
48. A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, number 24 in Algorithm and Combinatorics, Springer Verlag, 2003.
49. P. Seymour, Matroid minors, Handbook of Combinatorics, (Eds.: R. L. Graham, M. Grötschel and L. Lóvasz). North-Holland, Amsterdam, 1985, 419-431.
50. P. Seymour, Disjoint paths in Graphs, Discrete Mathematics 29 (1980), 293-309.
51. P. Seymour and R. Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory Ser. B 58 (1993), 22-33.
52. Y. Shiloach, A polynomial solution to the undirected two paths problem, J. ACM, 27 (1980), 445-456.
53. 4, J. Combin. Theory Ser. B 59 (1993), 74-84.
54. R.E. Tarjan, Data Structures and network algorithms, SIAM, Philadelphia, PA, 1983.
55. T. Tholey, Solving the 2-disjoint paths problem in nearly linear time, Theory of computing systems 39 (2004), 51-78.
56. A. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 261-265.
57. A. Thomason, The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001), 318-338.
58. C. Thomassen, 2-linked graph, European Journal of Combinatorics 1 (1980), 371-378.
59. C. Thomassen, On the presence of disjoint subgraphs of a specified type, Journal of Graph Theory, 12 (1988), 101-111.
60. 4-subdivision in graphs, in "Surveys in Combinatorics, 1999 (London)", London Math. Soc. Lecture Note Ser. 267, Cambridge Univ. Press, Cambridge, 1999, 223-238.
61. 4-subdivisions in 4-chromatic graphs. Combinatorica 21 (2001), 417-443. (Pubitemid 33584207)
62. B. Toft, Problems 10 and 11, in Recent advances in graph theory: Proceedings of the Symposium held in Prague, June 1974 (ed. M. Fielder), Academia Praha, Prague (1975), 543-544.
63. 4 is 3-colorable, J. Combin. Optimization, 2 (1998), 117-188. (Pubitemid 128690121)