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Published online by Cambridge University Press:  30 August 2017

Martin Grohe
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RWTH Aachen University, Germany
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References

[1] M., Ajtai and Y., Gurevich, Datalog vs. first-order logic, Proc. of the 30th Annual IEEE Symp. on Foundations of Computer Science, 1989, pp. 142–147.Google Scholar
[2] D., Archdeacon, A Kuratowski theorem for the projective plane, Journal of Graph Theory, vol. 5 (1981), pp. 243–246.Google Scholar
[3] S., Arnborg, D., Corneil, and A., Proskurowski, Complexity of finding embeddings in a k-tree, SIAM Journal on Algebraic Discrete Methods, vol. 8 (1987), pp. 277–284.Google Scholar
[4] S., Arnborg and A., Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Applied Mathematics, vol. 23 (1989), pp. 11–24.Google Scholar
[5] A., Atserias and E.N., Maneva, Graph isomorphism, Sherali-Adams relaxations and expressibility in counting logics, Electronic Colloquium on Computational Complexity (ECCC), vol. 18 (2011), no. 77.Google Scholar
[6] L., Babai, Moderately exponential bound for graph isomorphism, Fundamentals of Computation Theory, FCT'81(F., Gécseg, editor), Lecture Notes in Computer Science, vol. 117, Springer-Verlag, 1981, pp. 34–50.
[7] L., Babai, Graph isomorphism in quasipolynomial time, Proc. of the 48th Annual ACMSymp. on Theory of Computing (STOC '16), 2016, pp. 684–697.Google Scholar
[8] L., Babai, P., Erdös, and S., Selkow, Random graph isomorphism, SIAM Journal on Computing, vol. 9 (1980), pp. 628–635.Google Scholar
[9] L., Babai and E. M., Luks, Canonical labeling of graphs, Proc. of the 15th ACM Symp. on Theory of Computing, 1983, pp. 171–183.Google Scholar
[10] J., Barwise, On Moschovakis closure ordinals, The Journal of Symbolic Logic, vol. 42 (1977), pp. 292–296.Google Scholar
[11] J., Barwise and S., Feferman (editors), Model Theoretic Logics, Perspectives in Mathematical Logic, Springer-Verlag, 1985.
[12] C., Berkholz, P., Bonsma, and M., Grohe, Tight lower and upper bounds for the complexity of canonical colour refinement, Theory of Computing Systems, vol. doi:10.1007/s00224-016-9686-0 (2016).CrossRef
[13] H. L., Bodlaender, Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees, Journal of Algorithms, vol. 11 (1990), pp. 631–643.Google Scholar
[14] H. L., Bodlaender, A linear-time algorithm for finding tree-decompositions of small treewidth, SIAM Journal on Computing, vol. 25 (1996), pp. 1305–1317.Google Scholar
[15] R. B., Boppana, J., Hastad, and S., Zachos, Does co-NP have short interactive proofs?, Information Processing Letters, vol. 25 (1987), pp. 127–132.Google Scholar
[16] J., Cai, M., Fürer, and N., Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica, vol. 12 (1992), pp. 389–410.Google Scholar
[17] A., Cardon and M., Crochemore, Partitioning a graph in O (|A| log2 |V|), Theoretical Computer Science, vol. 19 (1982), no. 1, pp. 85–98.Google Scholar
[18] J., Carmesin, R., Diestel, F., Hundertmark, and M., Stein, Connectivity and tree structure in finite graphs, Combinatorica, vol. 34 (2014), no. 1, pp. 11–46.Google Scholar
[19] A., Chandra and D., Harel, Structure and complexity of relational queries, Journal of Computer and System Sciences, vol. 25 (1982), pp. 99–128.Google Scholar
[20] B., Courcelle and J., Engelfriet, Graph Structure and Monadic Second- Order Logic — A Language-Theoretic Approach, Cambridge University Press, 2012.
[21] B., Courcelle and S., Olariu, Upper bounds to the clique-width of graphs, Discrete Applied Mathematics, vol. 101 (2000), pp. 77–114.Google Scholar
[22] A., Dawar, Generalized quantifiers and logical reducibilities, Journal of Logic and Computation, vol. 5 (1995), pp. 213–226.Google Scholar
[23] A., Dawar, M., Grohe, and S., Kreutzer, Locally excluding a minor, Proc. of the 22nd IEEE Symp. on Logic in Computer Science, 2007, pp. 270–279.
[24] A., Dawar, M., Grohe, S., Kreutzer, and N., Schweikardt, Approximation schemes for first-order definable optimisation problems, Proc. of the 21st IEEE Symp. on Logic in Computer Science, 2006, pp. 411–420.Google Scholar
[25] A., Dawar, S., Lindell, and S., Weinstein, Infinitary logic and inductive definability over finite structures, Information and Computation, vol. 119 (1995), pp. 160–175.Google Scholar
[26] E. D., Demaine, F. V., Fomin,M. T., Hajiaghayi, and D. M., Thilikos, Subexponential parameterized algorithms on bounded-genus graphs and h-minorfree graphs, Journal of the ACM, vol. 52 (2005), no. 6, pp. 866–893.Google Scholar
[27] E.D., Demaine, M., Hajiaghayi, and K., Kawarabayashi, Approximation algorithms via structural results for apex-minor-free graphs, Proc. of the 36th International Colloquium on Automata, Languages and Programming, Part I(S., Albers, A., Marchetti-Spaccamela, Y., Matias, S.E., Nikoletseas, and W., Thomas, editors), Lecture Notes in Computer Science, vol. 5555, Springer- Verlag, 2009, pp. 316–327.
[28] E.D., Demaine, M. T., Hajiaghayi, and K., Kawarabayashi, Algorithmic graph minor theory: Decomposition, approximation, and coloring., Proc. of the 45th Annual IEEE Symp. on Foundations of Computer Science, 2005, pp. 637–646.Google Scholar
[29] R., Diestel, Graph Theory, 3rd ed., Springer-Verlag, 2005.
[30] H.-D., Ebbinghaus, Extended logics: The general framework, Model- Theoretic Logics (J., Barwise and S., Feferman, editors), Springer-Verlag, 1985, pp. 25–76.
[31] H.-D., Ebbinghaus and J., Flum, Finite Model Theory, 2nd ed., Springer- Verlag, 1999.
[32] H.-D., Ebbinghaus, J., Flum, and W., Thomas, Mathematical Logic, 2nd ed., Springer-Verlag, 1994.
[33] Z., Endemann,, 2012, Personal communication.
[34] R., Fagin, Generalized first-order spectra and polynomial-time recognizable sets, Complexity of Computation, SIAM-AMS Proc., Vol. 7 (R.M. Karp, editor), 1974, pp. 43–73.Google Scholar
[35] I. S., Filotti and J. N., Mayer, A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus, Proc. of the 12th ACM Symp. on Theory of Computing, 1980, pp. 236–243.Google Scholar
[36] L. S., Filotti, G. L., Miller, and J., Reif, On determining the genus of a graph in O(vO(g)) steps, Proc. of the 11th ACMSymp. on Theory of Computing, 1979, pp. 27–37.Google Scholar
[37] J., Flum and M., Grohe, On fixed-point logic with counting, The Journal of Symbolic Logic, vol. 65 (2000), no. 2, pp. 777–787.Google Scholar
[38] E., Grädel, Finite model theory and descriptive complexity, E., Grädel, P.G., Kolaitis, L., Libkin, M., Marx, J., Spencer, M.Y., Vardi, Y., Venema, and S., Weinstein, Finite Model Theory and Its Applications, Springer-Verlag, 2007, pp. 12–230.
[39] E., Grädel, P., G.|Kolaitis, L., Libkin, M., Marx, J., Spencer, M., Y.|Vardi, Y., Venema, and S., Weinstein, Finite Model Theory and Its Applications, Springer-Verlag, 2007.
[40] E., Grädel and M., Otto, Inductive definability with counting on finite structures, Computer Science Logic, 6th Workshop, CSL '92, San Miniato 1992, Selected Papers(E., Börger, G., Jäger, H. Kleine, Büning, S., Martini, and M.M., Richter, editors), Lecture Notes in Computer Science, vol. 702, Springer-Verlag, 1993, pp. 231–247.
[41] M., Grohe, Arity hierarchies, Annals of Pure and Applied Logic, vol. 82 (1996), no. 2, pp. 103–163.Google Scholar
[42] M., Grohe, Fixed-point logics on planar graphs, Proc. of the 13th IEEE Symp. on Logic in Computer Science, 1998, pp. 6–15.Google Scholar
[43] M., Grohe, Isomorphism testing for embeddable graphs through definability, Proc. of the 32nd ACM Symp. on Theory of Computing, 2000, pp. 63–72.Google Scholar
[44] M., Grohe, Local tree-width, excluded minors, and approximation algorithms, Combinatorica, vol. 23 (2003), no. 4, pp. 613–632.Google Scholar
[45] M., Grohe, Definable tree decompositions, Proc. of the 23rd IEEE Symp. on Logic in Computer Science, 2008, pp. 406–417.Google Scholar
[46] M., Grohe, Fixed-point definability and polynomial time on chordal graphs and line graphs, Fields of Logic and Computation: Essays Dedicated to Yuri Gurevich on the Occasion of His 70th Birthday (A., Blass, N., Dershowitz, and W., Reisig, editors), Lecture Notes in Computer Science, vol. 6300, Springer- Verlag, 2010, pp. 328–353.
[47] M., Grohe, Fixed-point definability and polynomial time on graphs with excluded minors, Proc. of the 25th IEEE Symp. on Logic in Computer Science, 2010, pp. 179–188.Google Scholar
[48] M., Grohe, From polynomial time queries to graph structure theory, Communications of the ACM, vol. 54 (2011), no. 6, pp. 104–112.Google Scholar
[49] M., Grohe, Fixed-point definability and polynomial time on graphs with excluded minors, Journal of the ACM, vol. 59 (2012), no. 5.Google Scholar
[50] M., Grohe, Quasi-4-connected components, Proc. of the 43rd International Colloquium on Automata, Languages and Programming (Track A)(I., Chatzigiannakis, M., Mitzenmacher, Y., Rabani, and D., Sangiorgi, editors), LIPIcs, vol. 55, SchlossDagstuhl –Leibniz-Zentrum f ür Informatik, 2016, pp. 8:1–8:13.
[51] M., Grohe, Tangled up in blue (a survey on connectivity, decompositions, and tangles), ArXiv (CoRR), vol. arXiv:1605.06704 [cs.DM] (2016).
[52] M., Grohe, B., Grußien, A., Hernich, and B., Laubner, L-recursion and a new logic for logarithmic space, Logical Methods in Computer Science, vol. 9 (2012).
[53] M., Grohe and J., Mariño, Definability and descriptive complexity on databases of bounded tree-width, Proc. of the 7th International Conference on Database Theory (C., Beeri and P., Buneman, editors), Lecture Notes in Computer Science, vol. 1540, Springer-Verlag, 1999, pp. 70–82.
[54] M., Grohe and D., Marx, Structure theorem and isomorphism test for graphs with excluded topological subgraphs, SIAM Journal on Computing, vol. 44 (2015), no. 1, pp. 114–159.Google Scholar
[55] M., Grohe and M., Otto, Pebble games and linear equations, Proc. of the 26th International Workshop on Computer Science Logic(P., Cégielski and A., Durand, editors), Leibniz International Proc. in Informatics (LIPIcs), vol. 16, 2012, pp. 289–304.
[56] M., Grohe and P., Schweitzer, Isomorphism testing for graphs of bounded rank width, Proc. of the 55th Annual IEEE Symp. on Foundations of Computer Science, 2015, pp. 1010–1029.Google Scholar
[57] J. L., Gross and T.W., Tucker, Topological Graph Theory,Wiley, 1987.
[58] B., Grussien, Isoperimetric inequalities on the hexagonal grid, ArXiv, vol. arXiv:1201.0697 [math.CO] (2012).
[59] Y., Gurevich, Toward logic tailored for computational complexity, Computation and Proof Theory (M. M., Richter, E., Börger, W., Oberschelp, B., Schinzel, and W., Thomas, editors), Lecture Notes in Mathematics, vol. 1104, Springer-Verlag, 1984, pp. 175–216.
[60] Y., Gurevich, Logic and the challenge of computer science, Current Trends in Theoretical Computer Science (E., Börger, editor), Computer Science Press, 1988, pp. 1–57.
[61] Y., Gurevich and S., Shelah, Fixed point extensions of first-order logic, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 265–280.Google Scholar
[62] R., Halin, S-Functions for graphs, Journal of Geometry, vol. 8 (1976), pp. 171–186.Google Scholar
[63] L., Hella, Logical hierarchies in PTIME, Information and Computation, vol. 129 (1996), pp. 1–19.Google Scholar
[64] L., Hella, Ph. G., Kolaitis, and K., Luosto, Almost everywhere equivalence of logics in finite model theory, The Bulletin of Symbolic Logic, vol. 2 (1996), pp. 422–443.Google Scholar
[65] W., Hildesheimer, Ich trage eine Eule nach Athen, Lieblose Legenden, Suhrkamp Verlag, 1962.
[66] J. E., Hopcroft and R., Tarjan, Isomorphism of planar graphs (working paper), Complexity of Computer Computations (R. E., Miller and J.W., Thatcher, editors), Plenum Press, 1972.
[67] J. E., Hopcroft and J. K., Wong, Linear time algorithm for isomorphism of planar graphs, Proc. of the 6th ACM Symp. on Theory of Computing, 1974, pp. 172–184.Google Scholar
[68] J.E., Hopcroft, An n log n algorithm for minimizing states in a finite automaton, Theory of Machines and Computations (Z., Kohavi and A., Paz, editors), Academic Press, 1971, pp. 189–196.
[69] N., Immerman, Number of quantifiers is better than number of tape cells, Journal of Computer and System Sciences, vol. 22 (1981), pp. 384–406.Google Scholar
[70] N., Immerman, Relational queries computable in polynomial time (extended abstract), Proc. of the 14thACMSymp. on Theory of Computing, 1982, pp. 147–152.Google Scholar
[71] N., Immerman, Upper and lower bounds for first-order expressibility, Journal of Computer and System Sciences, vol. 25 (1982), pp. 76–98.Google Scholar
[72] N., Immerman, Relational queries computable in polynomial time, Information and Control, vol. 68 (1986), pp. 86–104.Google Scholar
[73] N., Immerman, Expressibility as a complexity measure: results and directions, Proc. of the 2nd IEEE Symp. on Structure in Complexity Theory, 1987, pp. 194–202.Google Scholar
[74] N., Immerman, Languages that capture complexity classes, SIAM Journal on Computing, vol. 16 (1987), pp. 760–778.Google Scholar
[75] N., Immerman, Descriptive Complexity, Springer-Verlag, 1999.
[76] N., Immerman and E., Lander, Describing graphs: A first-order approach to graph canonization, Complexity Theory Retrospective (A., Selman, editor), Springer-Verlag, 1990, pp. 59–81.
[77] R. M., Karp, Reducibilities among combinatorial problems, Complexity of Computer Computations (R.E., Miller and J.W., Thatcher, editors), Plenum Press, New York, 1972, pp. 85–103.
[78] Ph. G., Kolaitis and M. Y., Vardi, Infinitary logic and 0-1 laws, Information and Computation, vol. 98 (1992), pp. 258–294.Google Scholar
[79] Ph. G., Kolaitis and M. Y., Vardi, On the expressive power of datalog: tools and a case study, Journal of Computer and System Sciences, vol. 51 (1995), no. 1, pp. 110–134.Google Scholar
[80] Ph. G., Kolaitis and M. Y., Vardi, On the expressive power of variable-confined logics, Proc. of the 26th IEEE Symp. on Logic in Computer Science, 1996, pp. 348–359.Google Scholar
[81] S., Kreutzer, Expressive equivalence of least and inflationary fixedpoint logic, Proc. of the 17th IEEE Symp. on Logic in Computer Science, 2002, pp. 403–413.Google Scholar
[82] K., Kuratowski, Sur le problème des courbes gauches en topologie, Fundamenta Mathematicae, vol. 15 (1930), pp. 271–283.Google Scholar
[83] B., Laubner, Capturing polynomial time on interval graphs, Proc. of the 25th IEEE Symp. on Logic in Computer Science, 2010, pp. 199–208.Google Scholar
[84] B., Laubner, The Structure of Graphs and New Logics for the Characterization of Polynomial Time, Ph.D. thesis, Humboldt-Universität zu Berlin, 2011.
[85] H. I., Levine, Homotopic curves on surfaces, Proc. of the American Mathematical Society, vol. 14 (1963), pp. 986–990.Google Scholar
[86] L., Libkin, Elements of Finite Model Theory, Springer-Verlag, 2004.
[87] E. M., Luks, Isomorphism of graphs of bounded valance can be tested in polynomial time, Journal of Computer and System Sciences, vol. 25 (1982), pp. 42–65.Google Scholar
[88] P., Malkin, Sherali–Adams relaxations of graph isomorphism polytopes, Discrete Optimization, vol. 12 (2014), pp. 73–97.Google Scholar
[89] G. L., Miller, Isomorphism testing for graphs of bounded genus, Proc. of the 12th ACM Symp. on Theory of Computing, 1980, pp. 225–235.Google Scholar
[90] G. L., Miller, Isomorphism of graphs which are pairwise k-separable, Information and Control, vol. 56 (1983), pp. 21–33.Google Scholar
[91] B., Mohar, A linear time algorithm for embedding graphs in an arbitrary surface, SIAM Journal on Discrete Mathematics, vol. 12 (1999), pp. 6–26.Google Scholar
[92] B., Mohar and C., Thomassen, Graphs on Surfaces, Johns Hopkins University Press, 2001.
[93] D., Neuen, Graph isomorphism for unit square graphs, Proc. of the 24th Annual European Symp. on Algorithms (ESA 2016) (P., Sankowski and Ch.D., Zaroliagis, editors), LIPIcs, vol. 57, Schloss Dagstuhl –Leibniz-Zentrum f ür Informatik, 2016, pp. 70:1–70:17.
[94] M., Otto, Bounded Variable Logics and Counting –A Study in Finite Models, Lecture Notes in Logic, vol. 9, Springer-Verlag, 1997.
[95] M., Otto, Capturing bisimulation-invariant Ptime., Proc. of the 4th International Symp. on Logical Foundations of Computer Science (S.I., Adian and A., Nerode, editors), Lecture Notes in Computer Science, vol. 1234, Springer-Verlag, 1997, pp. 294–305.
[96] S.-I., Oum and P. D., Seymour, Approximating clique-width and branchwidth, Journal of Combinatorial Theory, Series B, vol. 96 (2006), pp. 514–528.Google Scholar
[97] R., Paige and R. E., Tarjan, Three partition refinement algorithms, SIAM Journal on Computing, vol. 16 (1987), no. 6, pp. 973–989.Google Scholar
[98] L., Perkovic and B., Reed, An improved algorithm for finding tree decompositions of small width, International Journal of Foundations of Computer Science, vol. 11 (2000), no. 3, pp. 365–371.Google Scholar
[99] O., Pikhurko and O., Verbitsky, Logical complexity of graphs: a survey, Model Theoretic Methods in Finite Combinatorics (M., Grohe and J. A., Makowsky, editors), Contemporary Mathematics, vol. 558, American Mathematical Society, 2011, pp. 129–180.
[100] B., Poizat, Deux ou trois choses que je sais de Ln, The Journal of Symbolic Logic, vol. 47 (1982), pp. 641–658.Google Scholar
[101] I. N., Ponomarenko, The isomorphism problem for classes of graphs that are invariant with respect to contraction, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 174 (1988), no. Teor. Slozhn. Vychisl. 3, pp. 147–177, 182, In Russian.Google Scholar
[102] J., Redies, Defining PTIME Problems on Planar Graphs with few Variables, Master's thesis, Department of Compter Science, RWTH Aachen University, 2014.
[103] B., Reed, Finding approximate separators an computing tree-width quickly, Proc. of the 24th ACMSymp. on Theory of Computing, 1992, pp. 221–228.Google Scholar
[104] B., Reed, Tree width and tangles: A new connectivity measure and some applications, Surveys in Combinatorics (R. A., Bailey, editor), LMS Lecture Note Series, vol. 241, Cambridge University Press, 1997, pp. 87–162.
[105] N., Robertson and P. D., Seymour, Graph minors I–XXIII, Journal of Combinatorial Theory, Series B 1982–2012.
[106] N., Robertson and P. D., Seymour, Graph minors II. Algorithmic aspects of tree-width, Journal of Algorithms, vol. 7 (1986), pp. 309–322.Google Scholar
[107] N., Robertson and P. D., Seymour, Graph minors V. Excluding a planar graph, Journal of Combinatorial Theory, Series B, vol. 41 (1986), pp. 92–114.Google Scholar
[108] N., Robertson and P. D., Seymour, Graph minors IX. Disjoint crossed paths, Journal of Combinatorial Theory, Series B, vol. 49 (1990), pp. 40–77.Google Scholar
[109] N., Robertson and P. D., Seymour, Graph minors X. Obstructions to tree-decomposition, Journal of Combinatorial Theory, Series B, vol. 52 (1991), pp. 153–190.Google Scholar
[110] N., Robertson and P. D., Seymour, Graph minors XIII. The disjoint paths problem, Journal of Combinatorial Theory, Series B, vol. 63 (1995), pp. 65–110.Google Scholar
[111] N., Robertson and P. D., Seymour, Graph minors XVI. Excluding a non-planar graph, Journal of Combinatorial Theory, Series B, vol. 77 (1999), pp. 1–27.Google Scholar
[112] N., Robertson, P. D., Seymour, and R., Thomas, Linkless embeddings of graphs in 3-space, Bulletin of the AMS, vol. 28 (1993), pp. 84–89.Google Scholar
[113] N., Robertson and P.D., Seymour, Graph minors XX. Wagner's conjecture, Journal of Combinatorial Theory, Series B, vol. 92 (2004), pp. 325–357.Google Scholar
[114] N., Robertson and R., Vitray, Representativity of surface embeddings, Paths, Flows and VLSI-Layout (B., Korte, L., Lovász, H.J., Pr ömel, and A., Schrijver, editors), Springer-Verlag, 1990, pp. 293–328.
[115] U., Schöning, Graph isomorphism is in the low hierarchy, Journal of Computer and System Sciences, vol. 37 (1988), pp. 312–323.Google Scholar
[116] P., Schweitzer, Towards an isomorphism dichotomy for hereditary graph classes, Proc. of the 32nd International Symp. on Theoretical Aspects of Computer Science (E. W., Mayr and N., Ollinger, editors), LIPIcs, vol. 30, Schloss Dagstuhl –Leibniz-Zentrum f ür Informatik, 2015, pp. 689–702.
[117] D., Seese, Tree-partite graphs and the complexity of algorithms, Technical report, Akademie der Wissenschaften der DDR, Karl Weierstrass Institut f ür Mathematik, 1986.
[118] E., Selman, On fractional isomorphisms, 2013, Diplomarbeit at the Department of Mathematics, Humboldt-Universität zu Berlin.
[119] H. D., Sherali and W. P., Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM Journal on Discrete Mathematics, vol. 3 (1990), no. 3, pp. 411–430.Google Scholar
[120] C., Thomassen, The graph genus problem is NP-complete, Journal of Algorithms, vol. 10 (1988), pp. 458–576.Google Scholar
[121] C., Thomassen, The Jordan–Schönflies Theorem and the classification of surfaces, American Mathematical Monthly, vol. 99 (1992), pp. 116–130.Google Scholar
[122] G., Tinhofer, Anote on compact graphs, DiscreteApplied Mathematics, vol. 30 (1991), pp. 253–264.Google Scholar
[123] J., Torán, On the hardness of graph isomorphism, SIAM Journal on Computing, vol. 33 (2004), no. 5, pp. 1093–1108.Google Scholar
[124] R., Uehara, Simple geometrical intersection graphs, Proc. of the Second International Workshop on Algorithms and Computation, Lecture Notes in Computer Science, vol. 4921, Springer-Verlag, 2008, pp. 25–33.
[125] M. Y., Vardi, The complexity of relational query languages, Proc. of the 14th ACM Symp. on Theory of Computing, 1982, pp. 137–146.Google Scholar
[126] K., Wagner, Über eine Eigenschaft der ebenen Komplexe, Mathematische Annalen, vol. 114 (1937), pp. 570–590.Google Scholar
[127] K., Wagner, Beweis einer Abschwächung der Hadwiger-Vermutung, Mathematische Annalen, vol. 153 (1964), pp. 139–141.Google Scholar
[128] H., Whitney, Congruent graphs and the connectivity of graphs, American Journal of Mathematics, vol. 54 (1932), pp. 150–168.Google Scholar

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  • References
  • Martin Grohe, RWTH Aachen University, Germany
  • Book: Descriptive Complexity, Canonisation, and Definable Graph Structure Theory
  • Online publication: 30 August 2017
  • Chapter DOI: https://doi.org/10.1017/9781139028868.021
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  • References
  • Martin Grohe, RWTH Aachen University, Germany
  • Book: Descriptive Complexity, Canonisation, and Definable Graph Structure Theory
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  • Chapter DOI: https://doi.org/10.1017/9781139028868.021
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  • References
  • Martin Grohe, RWTH Aachen University, Germany
  • Book: Descriptive Complexity, Canonisation, and Definable Graph Structure Theory
  • Online publication: 30 August 2017
  • Chapter DOI: https://doi.org/10.1017/9781139028868.021
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