Skip to main content

Structural Properties of Sparse Graphs

  • Chapter
Building Bridges

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 19))

Abstract

In this chapter we briefly outline the main motivation of our work and we relate it to other research. We do not include any definition here.

Supported by grant 1M0021620808 of the Czech Ministry of Education and AEOLUS.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ajtai and Y. Gurevich, Monotone versus positive, Journal of the ACM, 34 (1987), 1004–1015.

    Article  MATH  MathSciNet  Google Scholar 

  2. N. Alechina and Y. Gurevich, Syntax vs. semantics on infinite structures, Structures in Logic and Computer Science (1997), 14–33.

    Google Scholar 

  3. N. Alon, P. D. Seymour and R. Thomas, A separator theorem for graphs with excluded minor and its applications, Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (1990), pp. 293–299.

    Google Scholar 

  4. N. Alon, P. D. Seymour and R. Thomas, A separator theorem for nonplanar graphs, Journal of the American Mathematical Society, 3 (1990), 801–808.

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Alon and A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, Proc. 46th IEEE FOCS (2005), pp. 429–438.

    Google Scholar 

  6. S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Applied Mathematics, 23 (1989), no. 1, 11–24.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Atserias, On digraph coloring problems and treewidth duality, 20th IEEE Symposium on Logic in Computer Science (LICS) (2005), pp. 106–115.

    Google Scholar 

  8. A. Atserias, A. Dawar and M. Grohe, Preservation under extensions on well-behaved finite structures, 32nd International Colloquium on Automata, Languages and Programming (ICALP) (Springer-Verlag, ed.), Lecture Notes in Computer Science, vol. 3580, 2005, pp. 1437–1449.

    Google Scholar 

  9. A. Atserias, A. Dawar and P. G. Kolaitis, On preservation under homomorphisms and unions of conjunctive queries, Proceedings of the twenty-third ACM SIGMODSIGACT-SIGART symposium on Principles of database systems, ACM Press, 2004, pp. 319–329.

    Google Scholar 

  10. A. Atserias, A. Dawar and P. G. Kolaitis, On preservation under homomorphisms and unions of conjunctive queries, Journal of the ACM, 53 (2006), 208–237.

    Article  MathSciNet  Google Scholar 

  11. H. L. Bodlaender, A linear-time algorithm for finding tree-decompositions of small treewidth, SIAM Journal of Computing, 25 (1996), no. 6, 1305–1317.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. L. Bodlaender, J. R. Gilbert, H. Hafsteinsson and T. Kloks, Approximating treewidth, pathwidth, frontsize and shortest elimination tree, Journal of Algorithms, 18 (1995), 238–255.

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Borgs, J. Chayes, L. Lovász, V. T. Sós and K. Vesztergombi, Counting graph homomorphisms, in: Topics in Discrete Mathematics (M. Klazar, J. Kratochvíl, M. Loebl, J. Matoušek, R. Thomas and P. Valtr, eds.), Algorithms and Combinatorics, vol. 26, Springer Verlag, 2006, (dedicated to Jarik Nešetřil on the Occasion of his 60th birthday), pp. 315–371.

    Google Scholar 

  14. O. V. Borodin, On acyclic colorings of planar graphs, Discrete Mathematics, 25 (1979), no. 3, 211–236.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Brandstädt, H.-O. Le and J.-M. Vanherpe, Structure and stability number of chair-, co-P-and gem-free graphs revisited, Information Processing Letters, 86 (2003), 161–167.

    Article  MATH  MathSciNet  Google Scholar 

  16. K. Cameron, Induced matchings, Discrete Applied Mathematics, 24 (1989), 97–102.

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Cameron, Induced matchings in intersection graphs, Discrete Mathematics, 278 (2004), 1–9.

    Article  MATH  MathSciNet  Google Scholar 

  18. K. Cameron, R. Sritharan and Y. Tang, Finding a maximum induced matching in weakly chordal graphs, Discrete Mathematics, 266 (2003), 133–142.

    Article  MATH  MathSciNet  Google Scholar 

  19. J.-M. Chang, Induced matchings in asteroidal triple-free graphs, Discrete Applied Mathematics, 132 (2004), 67–78.

    Article  Google Scholar 

  20. M. Chrobak and D. Eppstein, Planar orientations with low out-degree and compaction of adjacency matrices, Theoretical Computer Science, 86 (1991), 243–266.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. R. K. Chung, R. L. Graham and R. M. Wilson, Quasi-random graphs, Combinatorica, 9 (1989), no. 4, 345–362.

    Article  MATH  MathSciNet  Google Scholar 

  22. D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, J. Symbolic Comput., 9 (1990), 251–280.

    Article  MATH  MathSciNet  Google Scholar 

  23. B. Courcelle, Graph rewriting: an algebraic and logic approach, in: Handbook of Theoretical Computer Science (J. van Leeuwen, ed.), vol. 2, Elsevier, Amsterdam (1990), pp. 142–193.

    Google Scholar 

  24. B. Courcelle, The monadic second-order logic of graphs I: recognizable sets of finite graphs, Information Computation, 85 (1990), 12–75.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Dawar, Finite model theory on tame classes of structures, in: Mathematical Foundations of Computer Science 2007 (L. Kučera and A. Kučera, eds.), Lecture Notes in Computer Science, vol. 4708, Springer (2007), pp. 2–12.

    Google Scholar 

  26. A. Dawar, On preservation theorems in finite model theory, Invited talk at the 6th Panhellenic Logic Symposium — Volos, Greece, July 2007.

    Google Scholar 

  27. A. Dawar, M. Grohe and S. Kreutzer, Locally excluding a minor, Proc. 22nd IEEE Symp. on Logic in Computer Science (2007).

    Google Scholar 

  28. J.S. Deogun, T. Kloks, D. Kratsch and H. Muller, On vertex ranking for permutation and other graphs, in: Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science (P. Enjalbert, E. W. Mayr and K. W. Wagner, eds.), Lecture Notes in Computer Science, vol. 775, Springer, 1994, pp. 747–758.

    Google Scholar 

  29. M. DeVos, G. Ding, B. Oporowski, D. P. Sanders, B. Reed, P. D. Seymour and D. Vertigan, Excluding any graph as a minor allows a low tree-width 2-coloring, Journal of Combinatorial Theory, Series B, 91 (2004), 25–41.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Diestel, Graph theory, Springer Verlag (1997).

    MATH  Google Scholar 

  31. [31] R. G. Downey and M. R. Fellows, Parameterized computational feasibility, in: P. Clote, J. Remmel (eds.): Feasible Mathematics II, Boston: Birkhäuser (1995), pp. 219–244.

    Google Scholar 

  32. R. G. Downey and M. R. Fellows, Parameterized complexity, Springer (1999).

    Google Scholar 

  33. P. Dreyer, Ch. Malon and J. Nešetřil, Universal h-colorable graphs without a given configuration, Discrete Math., 250 (2002), 245–25.

    Article  MATH  MathSciNet  Google Scholar 

  34. W. Duckworth, N. C. Wormald and M. Zito, Maximum induced matchings of random cubic graphs, Journal of Computational and Applied Mathematics, 142 (2002), no. 1, 39–50.

    Article  MATH  MathSciNet  Google Scholar 

  35. Z. Dvořák, Asymptotical structure of combinatorial objects, Ph.D. thesis, Charles University, Faculty of Mathematics and Physics (2007).

    Google Scholar 

  36. Z. Dvořák, On forbidden subdivision characterization of graph classes, European Journal of Combinatorics (2007), (in press).

    Google Scholar 

  37. H.-D. Ebbinghaus and J. Flum, Finite model theory, Springer-Verlag (1996).

    Google Scholar 

  38. G. Elek and B. Szegedy, Limits of hypergraphs, removal and regularity lemmas, a non-standard approach, preprint.

    Google Scholar 

  39. J. Ellis, H. Fan and M. Fellows, The dominating set problem is fixed parameter tractable for graphs of bounded genus, J. Algorithms, 52 (2004), no. 2, 152–168.

    Article  MATH  MathSciNet  Google Scholar 

  40. D. Eppstein, Subgraph isomorphism in planar graphs and related problems, in: Proc. 6th Symp. Discrete Algorithms, ACM and SIAM (January 1995), pp. 632–640.

    Google Scholar 

  41. D. Eppstein, Subgraph isomorphism in planar graphs and related problems, Journal of Graph Algorithms & Applications, 3 (1999), no. 3, 1–27.

    MathSciNet  Google Scholar 

  42. D. Eppstein, Diameter and treewidth in minor-closed graph families, Algorithmica, 27 (2000), 275–291, Special issue on treewidth, graph minors and algorithms.

    Article  MATH  MathSciNet  Google Scholar 

  43. P. Erdős, Problems and results in combinatorial analysis and graph theory, Discrete Mathematics, 72 (1988), no. 1–3, 81–92.

    Article  MathSciNet  Google Scholar 

  44. U. Faigle, U. Kern, H. A. Kierstead and W. T. Trotter, On the game chromatic number of some classes of graphs, Ars Combinatoria, 35 (1993), 143–150.

    MATH  MathSciNet  Google Scholar 

  45. R. J. Faudree, A. Gyárfás, R. H. Schelp and Zs. Tuza, Induced matchings in bipartite graphs, Discrete Mathematics, 78 (1989), no. 1–2, 83–87.

    Article  MATH  MathSciNet  Google Scholar 

  46. G. Fricke and R. Laskar, Strong matchings in trees, Congressus Numerantium, 89 (1992), 239–244.

    MathSciNet  Google Scholar 

  47. M. R. Garey and D. S. Johnson, Computers and intractability; a guide to the theory of np-completeness, W. H. Freeman & Co., New York, NY, USA (1990).

    Google Scholar 

  48. J. R. Gilbert, J. P. Hutchinson and R. E. Tarjan, A separator theorem for graphs of bounded genus, J. Algorithms (1984), no. 5, 375–390.

    Google Scholar 

  49. M. C. Golumbic and M. C. Laskar, Irredundancy in circular arc graphs, Discrete Applied Mathematics, 44 (1993), 79–89.

    Article  MATH  MathSciNet  Google Scholar 

  50. M. C. Golumbic and M. Lewenstein, New results on induced matchings, Discrete Applied Mathematics, 101 (2000), no. 1–3, 157–165.

    Article  MATH  MathSciNet  Google Scholar 

  51. B. Grünbaum, Acyclic colorings of planar graphs, Israel Journal of Mathematics, 14 (1973), 390–408.

    Article  MATH  MathSciNet  Google Scholar 

  52. D. Guan and X. Zhu, The game chromatic number of outerplanar graphs, Journal of Graph Theory, 30 (1999), 67–70.

    Article  MATH  MathSciNet  Google Scholar 

  53. B. Guenin, Edge coloring plane regular multigraphs, manuscript.

    Google Scholar 

  54. Y. Gurevich, Toward logic tailored for computational complexity, Computation and Proof Theory (M. M. Richter et al., ed.), Lecture Notes in Mathematics, Springer-Verlag, 1984.

    Google Scholar 

  55. R. Häggkvist and P. Hell, Universality of A-mote graphs, Europ. J. Combinatorics, 14 (1993), 23–27.

    Article  MATH  Google Scholar 

  56. R. Halin, S-functions for graphs, Journal of Geometry, 8 (1976), 171–176.

    Article  MATH  MathSciNet  Google Scholar 

  57. P. Hell and J. Nešetřil, Graphs and homomorphisms, Oxford Lecture Series in Mathematics and its Applications, vol. 28, Oxford University Press (2004).

    Google Scholar 

  58. W. Hodges, Model theory, Cambridge University Press, 1993.

    Google Scholar 

  59. H. A. Kierstead, personal communication cited in [114].

    Google Scholar 

  60. H. A. Kierstead and D. Yang, Orderings on graphs and game coloring number, Order, 20 (2003), 255–264.

    Article  MATH  MathSciNet  Google Scholar 

  61. D. Kobler and U. Rotics, Finding maximum induced matchings in subclasses of claw-free and P 5-free graphs and in graphs with matching and induced matching of equal maximum size, Algorithmica, 37 (2003), 327–346.

    Article  MATH  MathSciNet  Google Scholar 

  62. Y. Kohayakawa and V. Rödl, Szemerédi’s regularity lemma and quasi-randomness, in: Recent Advances in Algorithmic Combinatorics (B. Reed and C. Linhares-Sales, eds.), CMS Books Math./Ouvrages Math. SMC, vol. 11, Springer, New-York (2003), pp. 289–347.

    Chapter  Google Scholar 

  63. J. Komlós and M. Simonovits, Szemerédi’s regularity lemma and its applications in graph theory, in: Combinatorics, Paul Erdős is Eighty, vol. 2, János Bolyai Math. Soc. (1993), pp. 295–352.

    Google Scholar 

  64. A. V. Kostochka and L. S. Melnikov, On bounds of the bisection width of cubic graphs, in: Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (J. Nesetril and M. Fiedler, eds.), Elsevier (1992), pp. 151–154.

    Google Scholar 

  65. M. Kreidler and D. Seese, Monadic NP and graph minors, in: Computer Science Logic, Lecture Notes in Computer Science, vol. 1584, Springer (1999), pp. 126–141.

    Google Scholar 

  66. B. Larose, C. Loten and C. Tardif, A characterisation of first-order constraint satisfaction problems, Logical Methods in Computer Science, 3 (2007), no. 4, 6, 22pp, (electronic).

    Article  MathSciNet  Google Scholar 

  67. L. Libkin, Elements of finite model theory, Springer-Verlag (2004).

    Google Scholar 

  68. R. Lipton and R. E. Tarjan, A separator theorem for planar graphs, SIAM Journal on Applied Mathematics, 36 (1979), no. 2, 177–189.

    Article  MATH  MathSciNet  Google Scholar 

  69. L. Lovász and B. Szegedy, Szemerédi lemma for the analyst, Geom. Func. Anal., 17 (2007), 252–270.

    Google Scholar 

  70. V. V. Lozin, On maximum induced matchings in bipartite graphs, Information Processing Letters, 81 (2002), 7–11.

    Article  MATH  MathSciNet  Google Scholar 

  71. V. V. Lozin and D. Rautenbach, Some results on graphs without long induced paths, Information Processing Letters, 88 (2003), 167–171.

    Article  MATH  MathSciNet  Google Scholar 

  72. G. L. Miller, S.-H. Teng, W. Thurston and S. A. Vavasis, Geometric separators for finite-element meshes, SIAM J. on Scientific Computing, 19 (1998), no. 2, 364–386.

    Article  MATH  MathSciNet  Google Scholar 

  73. R. Naserasr, Homomorphisms and edge-coloring of planar graphs, Journal of Combinatorial Theory, Series B (2005), to appear.

    Google Scholar 

  74. J. Nešetřil, Aspects of structural combinatorics, Taiwanese J. Math., 3 (1999), no. 4, 381–424.

    MATH  MathSciNet  Google Scholar 

  75. J. Nešetřil and P. Ossona de Mendez, Colorings and homomorphisms of minor closed classes, in: The Goodman-Pollack Festschrift (B. Aronov, S. Basu, J. Pach and M. Sharir, eds.), Algorithms and Combinatorics, vol. 25, Discrete & Computational Geometry (2003), pp. 651–664.

    Google Scholar 

  76. J. Nešetřil and P. Ossona de Mendez, How many colors may we require?, in: Prague Midsummer Combinatorial Workshop IX (M. Mareš, ed.), KAM-DIMATIA Series, no. 2004-686 (2004), abstract, pp. 27–30.

    Google Scholar 

  77. J. Nešetřil and P. Ossona de Mendez, Cuts and bounds, Discrete Mathematics, Structural Combinatorics — Combinatorial and Computational Aspects of Optimization, Topology and Algebra, 302 (2005), no. 1–3, 211–224.

    MATH  Google Scholar 

  78. J. Nešetřil and P. Ossona de Mendez, Folding, Journal of Combinatorial Theory, Series B, 96 (2006), no. 5, 730–739.

    Article  MATH  MathSciNet  Google Scholar 

  79. J. Nešetřil and P. Ossona de Mendez, Linear time low tree-width partitions and algorithmic consequences, in: STOC’06. Proceedings of the 38th Annual ACM Symposium on Theory of Computing, ACM Press (2006), pp. 391–400.

    Google Scholar 

  80. J. Nešetřil and P. Ossona de Mendez, Tree depth, subgraph coloring and homomorphism bounds, European Journal of Combinatorics, 27 (2006), no. 6, 1022–1041.

    Article  MATH  MathSciNet  Google Scholar 

  81. J. Nešetřil and P. Ossona de Mendez, Fraternal augmentations, arrangeability and linearly bounded Ramsey numbers, SIAM Journal on Discrete Mathematics (2007), submitted.

    Google Scholar 

  82. J. Nešetřil and P. Ossona de Mendez, Induced matchings and induced paths in graphs, Tech. Report 2007-810, KAM-DIMATIA Series, 2007.

    Google Scholar 

  83. J. Nešetřil and P. Ossona de Mendez, First order properties on nowhere dense structures, Journal of Symbolic Logic (2008), submitted.

    Google Scholar 

  84. J. Nešetřil and P. Ossona de Mendez, Grad and classes with bounded expansion I. decompositions, European Journal of Combinatorics, 29 (2008), no. 3, 760–776.

    Article  MATH  MathSciNet  Google Scholar 

  85. J. Nešetřil and P. Ossona de Mendez, Grad and classes with bounded expansion II. algorithmic aspects, European Journal of Combinatorics, 29 (2008), no. 3, 777–791.

    Article  MATH  MathSciNet  Google Scholar 

  86. J. Nešetřil and P. Ossona de Mendez, Grad and classes with bounded expansion III. restricted graph homomorphism dualities, European Journal of Combinatorics, 29 (2008), no. 4, 1012–1024.

    Article  MATH  MathSciNet  Google Scholar 

  87. J. Nešetřil and P. Ossona de Mendez, On nowhere dense graphs, European Journal of Combinatorics (2008), submitted.

    Google Scholar 

  88. J. Nešetřil and S. Poljak, Complexity of the subgraph problem, Comment. Math. Univ. Carol., 26.2 (1985), 415–420.

    Google Scholar 

  89. J. Nešetřil and A. Pultr, On classes of relations and graphs determined by subobjects and factorobjects, Discrete Mathematics, 22 (1978), 287–300.

    Article  MATH  MathSciNet  Google Scholar 

  90. J. Nešetřil and R. Šámal, Tension continuous maps-their structure and applications, European J. Comb., 29,4 (2008), 1025–1054.

    Google Scholar 

  91. J. Nešetřil and S. Shelah, Order of countable graphs, European J. Comb., 24 (2003), 649–663.

    Article  MATH  Google Scholar 

  92. J. Nešetřil and I. Švejdarová, Diameter of duals are linear, SIAM J. Discrete Math., 21,2 (2007), 374–384.

    Article  MATH  MathSciNet  Google Scholar 

  93. J. Nešetřil and C. Tardif, Duality theorems for finite structures (characterizing gaps and good characterizations), Journal of Combinatorial Theory, Series B, 80 (2000), 80–97.

    Article  MATH  MathSciNet  Google Scholar 

  94. J. Nešetřil and C. Tardif, Short answers to exponentially long questions: Extremal aspects of homomorphism duality, SIAM Journal of Discrete Mathematics, 19 (2005), no. 4, 914–920.

    Article  MATH  Google Scholar 

  95. S. Plotkin, S. Rao and W. D. Smith, Shallow excluded minors and improved graph decomposition, 5th Symp. Discrete Algorithms, SIAM (1994), 462–470.

    Google Scholar 

  96. N. Robertson and P. D. Seymour, Graph minors. I. Excluding a forest, Journal of Combinatorial Theory, Series B, 35 (1983), 39–61.

    Article  MATH  MathSciNet  Google Scholar 

  97. N. Robertson and P. D. Seymour, Graph minors — a survey, in: Surveys in Combinatorics (I. Anderson, ed.), Cambridge University Press (1985), pp. 153–171.

    Google Scholar 

  98. N. Robertson and P. D. Seymour, Graph Minors. II. Algorithmic aspects of tree-width, Journal of Algorithms, 7 (1986), 309–322.

    Article  MATH  MathSciNet  Google Scholar 

  99. N. Robertson and P. D. Seymour, Graph Minors. VIII., Journal of Combinatorial Theory, Series B, 48 (1990), no. 2, 227–254.

    Article  MATH  MathSciNet  Google Scholar 

  100. B. Rossman, Homomorphisms and first-order logic, Journal of the ACM (2007), submitted.

    Google Scholar 

  101. P. Schaffer, Optimal node ranking of trees in linear time, Information Processing Letters, 33 (1989/90), 91–96.

    Article  MathSciNet  Google Scholar 

  102. A. Steger and M.-L. Yu, On induced matchings, Discrete Mathematics, 120 (1993), no. 1–3, 291–295.

    Article  MATH  MathSciNet  Google Scholar 

  103. L. J. Stockmeyer and V. V. Vazirani, NP-completeness of some generalizations of the maximum matching problem, Information Processing Letters, 15 (1982), no. 1, 14–19.

    Article  MATH  MathSciNet  Google Scholar 

  104. A. Stolboushkin, Finite monotone properties, in: Proc. 10th IEEE Symp. on Logic in Computer Science (1995), pp. 324–330.

    Google Scholar 

  105. L. Sunil Chandran, A high girth graph construction, SIAM J. Discret. Math., 16 (2003), no. 3, 366–370.

    Google Scholar 

  106. E. Szemerédi, Regular partitions of graphs, Problémes combinatoires et théorie des graphes, CNRS, 1976, pp. 399–401.

    Google Scholar 

  107. W. Tait, A counterexample to a conjecture of Scott and Suppes, Journal of Symbolic Logic, 24 (1959), 15–16.

    Article  MATH  MathSciNet  Google Scholar 

  108. T. Tao, The dichotomy between structure and randomness, arithmetic progression and the primes, in: Proceedings of the International Congress of Mathematicians (Madrid 2006) (European Math. Society, ed.), vol. 1 (2007), pp. 581–608.

    Google Scholar 

  109. S.-H. Teng, Combinatorial aspects of geometric graphs, Computational Geometry (1998), no. 9, 277–287.

    Google Scholar 

  110. C. Thomassen, A short list color proof of Grötzsch theorem, Journal of Combinatorial Theory, Series B, 88 (2003), 189–192.

    Article  MATH  MathSciNet  Google Scholar 

  111. K. Wagner, Über eine Eigenschaft der Ebenen Komplexe, Mathematische Annalen, 114 (1937), 570–590.

    Google Scholar 

  112. R. Yuster, Combinatorial and computational aspects of graph packing and graph decompositions, Computer Science Review, 1 (2007), no. 1, 12–26.

    Article  Google Scholar 

  113. X. Zhu, The game coloring number of planar graphs, Journal of Combinatorial Theory, Series B, 75 (1999), no. 2, 245–258.

    Article  MATH  MathSciNet  Google Scholar 

  114. X. Zhu, The game coloring number of pseudo partial k-trees, Discrete Mathematics, 215 (2000), 245–262.

    Article  MATH  MathSciNet  Google Scholar 

  115. X. Zhu, Refined activation strategy for the marking game, Journal of Combinatorial Theory, Series B (2007), in press.

    Google Scholar 

  116. X. Zhu, Colouring graphs with bounded generalized colouring number, Discrete Mathematics (2008), in press.

    Google Scholar 

  117. M. Zito, Maximum induced matchings in regular graphs and trees, in: Proceedings of WG’ 99: the 25th International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 1665 (1999), pp. 89–100.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

NeŠetřil, J., De Mendez, P.O. (2008). Structural Properties of Sparse Graphs. In: Grötschel, M., Katona, G.O.H., Sági, G. (eds) Building Bridges. Bolyai Society Mathematical Studies, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85221-6_13

Download citation

Publish with us

Policies and ethics