Logic of graphs (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Logic of graphs" in English language version.

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Spencer (2001), Section 1.2, "What Is a First Order Theory?", pp. 15–17. Spencer, Joel (2001), The Strange Logic of Random Graphs, Algorithms and Combinatorics, vol. 22, Springer-Verlag, Berlin, doi:10.1007/978-3-662-04538-1, ISBN 3-540-41654-4, MR 1847951
Verbitsky & Zhukovskii (2019). Verbitsky, Oleg; Zhukovskii, Maksim (2019), "Tight bounds on the asymptotic descriptive complexity of subgraph isomorphism", ACM Transactions on Computational Logic, 20 (2): A9:1–A9:18, arXiv:1802.02143, doi:10.1145/3303881, MR 3942556, S2CID 3603039
Zeume (2017). Zeume, Thomas (2017), "The dynamic descriptive complexity of $k$ -clique" (PDF), Information and Computation, 256: 9–22, doi:10.1016/j.ic.2017.04.005, MR 3705411, S2CID 1412001
For instance, Henson (1972) requires directed graphs to be described by an asymmetric relation, meaning that loops and 2-cycles are both disallowed, giving the oriented graphs. Henson, C. Ward (1972), "Countable homogeneous relational structures and $\aleph _{0}$ -categorical theories", The Journal of Symbolic Logic, 37: 494–500, doi:10.2307/2272734, JSTOR 2272734, MR 0321727, S2CID 40662635
Koncewicz (1973). Koncewicz, Leszek (1973), "Definability of classes of graphs in the first order predicate calculus with identity", Polish Academy of Sciences, 32: 159–190, doi:10.1007/BF02123839, JSTOR 20014678, MR 0351796, S2CID 189786935
Glebskiĭ et al. (1969); Fagin (1976) Glebskiĭ, Ju. V.; Kogan, D. I.; Liogon'kiĭ, M. I.; Talanov, V. A. (1969), "Volume and fraction of satisfiability of formulas of the lower predicate calculus", Otdelenie Matematiki, Mekhaniki i Kibernetiki Akademii Nauk Ukrainskoĭ SSR: Kibernetika (2): 17–27, MR 0300882 Fagin, Ronald (1976), "Probabilities on finite models", Journal of Symbolic Logic, 41 (1): 50–58, doi:10.1017/s0022481200051756, JSTOR 2272945, MR 0476480, S2CID 2563318
Grandjean (1983). Grandjean, Étienne (1983), "Complexity of the first-order theory of almost all finite structures", Information and Control, 57 (2–3): 180–204, doi:10.1016/S0019-9958(83)80043-6, MR 0742707
Shelah & Spencer (1988); Spencer (2001). Shelah, Saharon; Spencer, Joel (1988), "Zero-one laws for sparse random graphs", Journal of the American Mathematical Society, 1 (1): 97–115, doi:10.2307/1990968, JSTOR 1990968, MR 0924703 Spencer, Joel (2001), The Strange Logic of Random Graphs, Algorithms and Combinatorics, vol. 22, Springer-Verlag, Berlin, doi:10.1007/978-3-662-04538-1, ISBN 3-540-41654-4, MR 1847951
Lynch (1992). Lynch, James F. (1992), "Probabilities of sentences about very sparse random graphs", Random Structures & Algorithms, 3 (1): 33–53, doi:10.1002/rsa.3240030105, MR 1139487
Spencer (1990). Spencer, Joel (1990), "Infinite spectra in the first order theory of graphs", Combinatorica, 10 (1): 95–102, doi:10.1007/BF02122699, MR 1075070, S2CID 27770505
Nešetřil & Ossona de Mendez (2012), 18.3 The Subgraph Isomorphism Problem and Boolean Queries, pp. 400–401; Dvořák, Kráľ & Thomas (2010); Grohe, Kreutzer & Siebertz (2014). Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer-Verlag, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058 Dvořák, Zdeněk; Kráľ, Daniel; Thomas, Robin (2010), "Deciding first-order properties for sparse graphs", Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 133–142, CiteSeerX 10.1.1.170.9781, doi:10.1109/FOCS.2010.20, ISBN 978-0-7695-4244-7, MR 3024787, S2CID 15264036 Grohe, Martin; Kreutzer, Stephan; Siebertz, Sebastian (2014), "Deciding first-order properties of nowhere dense graphs", Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC '14), New York: ACM, pp. 89–98, arXiv:1311.3899, doi:10.1145/2591796.2591851, ISBN 978-1-4503-2710-7, S2CID 13297133
Pikhurko, Spencer & Verbitsky (2006). Pikhurko, Oleg; Spencer, Joel; Verbitsky, Oleg (2006), "Succinct definitions in the first order theory of graphs", Annals of Pure and Applied Logic, 139 (1–3): 74–109, arXiv:math/0401307, doi:10.1016/j.apal.2005.04.003, MR 2206252, S2CID 3041191
Verbitsky (2005). Verbitsky, Oleg (2005), "The first order definability of graphs with separators via the Ehrenfeucht game", Theoretical Computer Science, 343 (1–2): 158–176, arXiv:math/0401361, doi:10.1016/j.tcs.2005.05.003, MR 2168849, S2CID 17886484
Immerman & Lander (1990). Immerman, Neil; Lander, Eric (1990), "Describing graphs: a first-order approach to graph canonization", in Selman, Alan L. (ed.), Complexity Theory Retrospective: In honor of Juris Hartmanis on the occasion of his sixtieth birthday, New York: Springer-Verlag, pp. 59–81, doi:10.1007/978-1-4612-4478-3_5, MR 1060782
Ebbinghaus & Flum (1995). Parys (2014) writes that this undecidability result is well known, and attributes it to Trahtenbrot (1950) on the undecidability of first-order satisfiability for more general classes of finite structures. Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995), Finite Model Theory, Springer Monographs in Mathematics (2nd ed.), Springer, p. 129, doi:10.1007/3-540-28788-4 Parys, Paweł (2014), "First-order logic on CPDA graphs", Computer science—theory and applications, Lecture Notes in Computer Science, vol. 8476, New York: Springer-Verlag, pp. 300–313, doi:10.1007/978-3-319-06686-8_23, MR 3218557, S2CID 31640587 Trahtenbrot, B. A. (1950), "The impossibility of an algorithm for the decision problem for finite domains", Doklady Akademii Nauk SSSR, New Series, 70: 569–572, MR 0033784
Lavrov (1963). Lavrov, I. A. (1963), "The effective non-separability of the set of identically true formulae and the set of finitely refutable formulae for certain elementary theories", Algebra i Logika Sem., 2 (1): 5–18, MR 0157904
Grohe (2017), pp. 23–27. Grohe, Martin (2017), Descriptive complexity, canonisation, and definable graph structure theory, Lecture Notes in Logic, vol. 47, Cambridge University Press, Cambridge, ISBN 978-1-107-01452-7, MR 3729479
Grohe (2017), pp. 50–51. Grohe, Martin (2017), Descriptive complexity, canonisation, and definable graph structure theory, Lecture Notes in Logic, vol. 47, Cambridge University Press, Cambridge, ISBN 978-1-107-01452-7, MR 3729479
Cai, Fürer & Immerman (1992). Cai, Jin-Yi; Fürer, Martin; Immerman, Neil (1992), "An optimal lower bound on the number of variables for graph identification", Combinatorica, 12 (4): 389–410, doi:10.1007/BF01305232, MR 1194730
Hella, Kolaitis & Luosto (1996). Hella, Lauri; Kolaitis, Phokion G.; Luosto, Kerkko (1996), "Almost everywhere equivalence of logics in finite model theory", The Bulletin of Symbolic Logic, 2 (4): 422–443, doi:10.2307/421173, JSTOR 421173, MR 1460316, S2CID 16411368
Laubner (2010). Laubner, Bastian (2010), "Capturing polynomial time on interval graphs", 25th Annual IEEE Symposium on Logic in Computer Science (LICS 2010), Los Alamitos, California: IEEE Computer Society, pp. 199–208, arXiv:0911.3799, doi:10.1109/LICS.2010.42, MR 2963094, S2CID 1450409
These definitions can be found e.g. in Courcelle & Engelfriet (2012), p. 69, under the slightly different notation MS₁ and MS₂. Other authors used the MSO₁ and MSO₂ notations, and Courcelle came to use it later as well; see, e.g., Courcelle (2018). Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Courcelle, Bruno (2018), "Fly-automata for checking MSO₂ graph properties", Discrete Applied Mathematics, 245: 236–252, arXiv:1511.08605, doi:10.1016/j.dam.2016.10.018, MR 3804787
Fagin, Stockmeyer & Vardi (1995). Fagin, Ronald; Stockmeyer, Larry J.; Vardi, Moshe Y. (1995), "On monadic NP vs monadic co-NP", Information and Computation, 120 (1): 78–92, doi:10.1006/inco.1995.1100, MR 1340807
Courcelle & Engelfriet (2012); Libkin (2004), Corollary 7.24, pp. 126–127. Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Libkin, Leonid (2004), Elements of finite model theory, Texts in Theoretical Computer Science: An EATCS Series, Springer-Verlag, Berlin, doi:10.1007/978-3-662-07003-1, ISBN 3-540-21202-7, MR 2102513, S2CID 30176939
Courcelle (1996). Courcelle, Bruno (1996), "On the expression of graph properties in some fragments of monadic second-order logic" (PDF), in Immerman, Neil; Kolaitis, Phokion G. (eds.), Proc. Descr. Complex. Finite Models, DIMACS, vol. 31, Amer. Math. Soc., pp. 33–62, CiteSeerX 10.1.1.55.5184, MR 1451381
Courcelle & Oum (2007). Courcelle, Bruno; Oum, Sang-il (2007), "Vertex-minors, monadic second-order logic, and a conjecture by Seese" (PDF), Journal of Combinatorial Theory, Series B, 97 (1): 91–126, doi:10.1016/j.jctb.2006.04.003, MR 2278126
Seese (1991). Seese, D. (1991), "The structure of the models of decidable monadic theories of graphs", Annals of Pure and Applied Logic, 53 (2): 169–195, doi:10.1016/0168-0072(91)90054-P, MR 1114848

archive.org

scholar.archive.org

Glebskiĭ et al. (1969); Fagin (1976) Glebskiĭ, Ju. V.; Kogan, D. I.; Liogon'kiĭ, M. I.; Talanov, V. A. (1969), "Volume and fraction of satisfiability of formulas of the lower predicate calculus", Otdelenie Matematiki, Mekhaniki i Kibernetiki Akademii Nauk Ukrainskoĭ SSR: Kibernetika (2): 17–27, MR 0300882 Fagin, Ronald (1976), "Probabilities on finite models", Journal of Symbolic Logic, 41 (1): 50–58, doi:10.1017/s0022481200051756, JSTOR 2272945, MR 0476480, S2CID 2563318

archive.org

Courcelle & Engelfriet (2012); Libkin (2004), Corollary 7.24, pp. 126–127. Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Libkin, Leonid (2004), Elements of finite model theory, Texts in Theoretical Computer Science: An EATCS Series, Springer-Verlag, Berlin, doi:10.1007/978-3-662-07003-1, ISBN 3-540-21202-7, MR 2102513, S2CID 30176939

arxiv.org

Verbitsky & Zhukovskii (2019). Verbitsky, Oleg; Zhukovskii, Maksim (2019), "Tight bounds on the asymptotic descriptive complexity of subgraph isomorphism", ACM Transactions on Computational Logic, 20 (2): A9:1–A9:18, arXiv:1802.02143, doi:10.1145/3303881, MR 3942556, S2CID 3603039
Nešetřil & Ossona de Mendez (2012), 18.3 The Subgraph Isomorphism Problem and Boolean Queries, pp. 400–401; Dvořák, Kráľ & Thomas (2010); Grohe, Kreutzer & Siebertz (2014). Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer-Verlag, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058 Dvořák, Zdeněk; Kráľ, Daniel; Thomas, Robin (2010), "Deciding first-order properties for sparse graphs", Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 133–142, CiteSeerX 10.1.1.170.9781, doi:10.1109/FOCS.2010.20, ISBN 978-0-7695-4244-7, MR 3024787, S2CID 15264036 Grohe, Martin; Kreutzer, Stephan; Siebertz, Sebastian (2014), "Deciding first-order properties of nowhere dense graphs", Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC '14), New York: ACM, pp. 89–98, arXiv:1311.3899, doi:10.1145/2591796.2591851, ISBN 978-1-4503-2710-7, S2CID 13297133
Pikhurko, Spencer & Verbitsky (2006). Pikhurko, Oleg; Spencer, Joel; Verbitsky, Oleg (2006), "Succinct definitions in the first order theory of graphs", Annals of Pure and Applied Logic, 139 (1–3): 74–109, arXiv:math/0401307, doi:10.1016/j.apal.2005.04.003, MR 2206252, S2CID 3041191
Pikhurko & Verbitsky (2011). Pikhurko, Oleg; Verbitsky, Oleg (2011), "Logical complexity of graphs: a survey", in Grohe, Martin; Makowsky, Johann A. (eds.), Model Theoretic Methods in Finite Combinatorics (AMS-ASL Joint Special Session, January 5-8, 2009, Washington, DC), Contemporary Mathematics, vol. 558, American Mathematical Society, pp. 129–180, arXiv:1003.4865, ISBN 978-0-8218-8322-8
Verbitsky (2005). Verbitsky, Oleg (2005), "The first order definability of graphs with separators via the Ehrenfeucht game", Theoretical Computer Science, 343 (1–2): 158–176, arXiv:math/0401361, doi:10.1016/j.tcs.2005.05.003, MR 2168849, S2CID 17886484
Laubner (2010). Laubner, Bastian (2010), "Capturing polynomial time on interval graphs", 25th Annual IEEE Symposium on Logic in Computer Science (LICS 2010), Los Alamitos, California: IEEE Computer Society, pp. 199–208, arXiv:0911.3799, doi:10.1109/LICS.2010.42, MR 2963094, S2CID 1450409
These definitions can be found e.g. in Courcelle & Engelfriet (2012), p. 69, under the slightly different notation MS₁ and MS₂. Other authors used the MSO₁ and MSO₂ notations, and Courcelle came to use it later as well; see, e.g., Courcelle (2018). Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Courcelle, Bruno (2018), "Fly-automata for checking MSO₂ graph properties", Discrete Applied Mathematics, 245: 236–252, arXiv:1511.08605, doi:10.1016/j.dam.2016.10.018, MR 3804787
Grohe (2001); Kawarabayashi & Reed (2007). Grohe, Martin (2001), "Computing crossing numbers in quadratic time", Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (STOC '01), pp. 231–236, arXiv:cs/0009010, doi:10.1145/380752.380805, S2CID 724544 Kawarabayashi, Ken-ichi; Reed, Bruce (2007), "Computing crossing number in linear time", Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing (STOC '07), pp. 382–390, doi:10.1145/1250790.1250848, S2CID 13000831

books.google.com

Spencer (2001), Section 1.2, "What Is a First Order Theory?", pp. 15–17. Spencer, Joel (2001), The Strange Logic of Random Graphs, Algorithms and Combinatorics, vol. 22, Springer-Verlag, Berlin, doi:10.1007/978-3-662-04538-1, ISBN 3-540-41654-4, MR 1847951
Pikhurko & Verbitsky (2011). Pikhurko, Oleg; Verbitsky, Oleg (2011), "Logical complexity of graphs: a survey", in Grohe, Martin; Makowsky, Johann A. (eds.), Model Theoretic Methods in Finite Combinatorics (AMS-ASL Joint Special Session, January 5-8, 2009, Washington, DC), Contemporary Mathematics, vol. 558, American Mathematical Society, pp. 129–180, arXiv:1003.4865, ISBN 978-0-8218-8322-8
Courcelle & Engelfriet (2012); Libkin (2004), Corollary 7.24, pp. 126–127. Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Libkin, Leonid (2004), Elements of finite model theory, Texts in Theoretical Computer Science: An EATCS Series, Springer-Verlag, Berlin, doi:10.1007/978-3-662-07003-1, ISBN 3-540-21202-7, MR 2102513, S2CID 30176939

doi.org

Spencer (2001), Section 1.2, "What Is a First Order Theory?", pp. 15–17. Spencer, Joel (2001), The Strange Logic of Random Graphs, Algorithms and Combinatorics, vol. 22, Springer-Verlag, Berlin, doi:10.1007/978-3-662-04538-1, ISBN 3-540-41654-4, MR 1847951
Verbitsky & Zhukovskii (2019). Verbitsky, Oleg; Zhukovskii, Maksim (2019), "Tight bounds on the asymptotic descriptive complexity of subgraph isomorphism", ACM Transactions on Computational Logic, 20 (2): A9:1–A9:18, arXiv:1802.02143, doi:10.1145/3303881, MR 3942556, S2CID 3603039
Zeume (2017). Zeume, Thomas (2017), "The dynamic descriptive complexity of $k$ -clique" (PDF), Information and Computation, 256: 9–22, doi:10.1016/j.ic.2017.04.005, MR 3705411, S2CID 1412001
Goldberg (1993). Goldberg, Leslie Ann (1993), "Polynomial space polynomial delay algorithms for listing families of graphs", Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing (STOC '93), New York, NY, USA: ACM, pp. 218–225, doi:10.1145/167088.167160, ISBN 0-89791-591-7, S2CID 6305108
For instance, Henson (1972) requires directed graphs to be described by an asymmetric relation, meaning that loops and 2-cycles are both disallowed, giving the oriented graphs. Henson, C. Ward (1972), "Countable homogeneous relational structures and $\aleph _{0}$ -categorical theories", The Journal of Symbolic Logic, 37: 494–500, doi:10.2307/2272734, JSTOR 2272734, MR 0321727, S2CID 40662635
Koncewicz (1973). Koncewicz, Leszek (1973), "Definability of classes of graphs in the first order predicate calculus with identity", Polish Academy of Sciences, 32: 159–190, doi:10.1007/BF02123839, JSTOR 20014678, MR 0351796, S2CID 189786935
Glebskiĭ et al. (1969); Fagin (1976) Glebskiĭ, Ju. V.; Kogan, D. I.; Liogon'kiĭ, M. I.; Talanov, V. A. (1969), "Volume and fraction of satisfiability of formulas of the lower predicate calculus", Otdelenie Matematiki, Mekhaniki i Kibernetiki Akademii Nauk Ukrainskoĭ SSR: Kibernetika (2): 17–27, MR 0300882 Fagin, Ronald (1976), "Probabilities on finite models", Journal of Symbolic Logic, 41 (1): 50–58, doi:10.1017/s0022481200051756, JSTOR 2272945, MR 0476480, S2CID 2563318
Grandjean (1983). Grandjean, Étienne (1983), "Complexity of the first-order theory of almost all finite structures", Information and Control, 57 (2–3): 180–204, doi:10.1016/S0019-9958(83)80043-6, MR 0742707
Shelah & Spencer (1988); Spencer (2001). Shelah, Saharon; Spencer, Joel (1988), "Zero-one laws for sparse random graphs", Journal of the American Mathematical Society, 1 (1): 97–115, doi:10.2307/1990968, JSTOR 1990968, MR 0924703 Spencer, Joel (2001), The Strange Logic of Random Graphs, Algorithms and Combinatorics, vol. 22, Springer-Verlag, Berlin, doi:10.1007/978-3-662-04538-1, ISBN 3-540-41654-4, MR 1847951
Lynch (1992). Lynch, James F. (1992), "Probabilities of sentences about very sparse random graphs", Random Structures & Algorithms, 3 (1): 33–53, doi:10.1002/rsa.3240030105, MR 1139487
Spencer (1990). Spencer, Joel (1990), "Infinite spectra in the first order theory of graphs", Combinatorica, 10 (1): 95–102, doi:10.1007/BF02122699, MR 1075070, S2CID 27770505
Downey & Fellows (1995). Downey, R. G.; Fellows, M. R. (1995), "Fixed-parameter tractability and completeness. II. On completeness for W[1]", Theoretical Computer Science, 141 (1–2): 109–131, doi:10.1016/0304-3975(94)00097-3
Chen et al. (2006). Chen, Jianer; Huang, Xiuzhen; Kanj, Iyad A.; Xia, Ge (2006), "Strong computational lower bounds via parameterized complexity", Journal of Computer and System Sciences, 72 (8): 1346–1367, doi:10.1016/j.jcss.2006.04.007
Nešetřil & Ossona de Mendez (2012), 18.3 The Subgraph Isomorphism Problem and Boolean Queries, pp. 400–401; Dvořák, Kráľ & Thomas (2010); Grohe, Kreutzer & Siebertz (2014). Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer-Verlag, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058 Dvořák, Zdeněk; Kráľ, Daniel; Thomas, Robin (2010), "Deciding first-order properties for sparse graphs", Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 133–142, CiteSeerX 10.1.1.170.9781, doi:10.1109/FOCS.2010.20, ISBN 978-0-7695-4244-7, MR 3024787, S2CID 15264036 Grohe, Martin; Kreutzer, Stephan; Siebertz, Sebastian (2014), "Deciding first-order properties of nowhere dense graphs", Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC '14), New York: ACM, pp. 89–98, arXiv:1311.3899, doi:10.1145/2591796.2591851, ISBN 978-1-4503-2710-7, S2CID 13297133
Pikhurko, Spencer & Verbitsky (2006). Pikhurko, Oleg; Spencer, Joel; Verbitsky, Oleg (2006), "Succinct definitions in the first order theory of graphs", Annals of Pure and Applied Logic, 139 (1–3): 74–109, arXiv:math/0401307, doi:10.1016/j.apal.2005.04.003, MR 2206252, S2CID 3041191
Verbitsky (2005). Verbitsky, Oleg (2005), "The first order definability of graphs with separators via the Ehrenfeucht game", Theoretical Computer Science, 343 (1–2): 158–176, arXiv:math/0401361, doi:10.1016/j.tcs.2005.05.003, MR 2168849, S2CID 17886484
Immerman & Lander (1990). Immerman, Neil; Lander, Eric (1990), "Describing graphs: a first-order approach to graph canonization", in Selman, Alan L. (ed.), Complexity Theory Retrospective: In honor of Juris Hartmanis on the occasion of his sixtieth birthday, New York: Springer-Verlag, pp. 59–81, doi:10.1007/978-1-4612-4478-3_5, MR 1060782
Ebbinghaus & Flum (1995). Parys (2014) writes that this undecidability result is well known, and attributes it to Trahtenbrot (1950) on the undecidability of first-order satisfiability for more general classes of finite structures. Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995), Finite Model Theory, Springer Monographs in Mathematics (2nd ed.), Springer, p. 129, doi:10.1007/3-540-28788-4 Parys, Paweł (2014), "First-order logic on CPDA graphs", Computer science—theory and applications, Lecture Notes in Computer Science, vol. 8476, New York: Springer-Verlag, pp. 300–313, doi:10.1007/978-3-319-06686-8_23, MR 3218557, S2CID 31640587 Trahtenbrot, B. A. (1950), "The impossibility of an algorithm for the decision problem for finite domains", Doklady Akademii Nauk SSSR, New Series, 70: 569–572, MR 0033784
Cai, Fürer & Immerman (1992). Cai, Jin-Yi; Fürer, Martin; Immerman, Neil (1992), "An optimal lower bound on the number of variables for graph identification", Combinatorica, 12 (4): 389–410, doi:10.1007/BF01305232, MR 1194730
Hella, Kolaitis & Luosto (1996). Hella, Lauri; Kolaitis, Phokion G.; Luosto, Kerkko (1996), "Almost everywhere equivalence of logics in finite model theory", The Bulletin of Symbolic Logic, 2 (4): 422–443, doi:10.2307/421173, JSTOR 421173, MR 1460316, S2CID 16411368
Laubner (2010). Laubner, Bastian (2010), "Capturing polynomial time on interval graphs", 25th Annual IEEE Symposium on Logic in Computer Science (LICS 2010), Los Alamitos, California: IEEE Computer Society, pp. 199–208, arXiv:0911.3799, doi:10.1109/LICS.2010.42, MR 2963094, S2CID 1450409
These definitions can be found e.g. in Courcelle & Engelfriet (2012), p. 69, under the slightly different notation MS₁ and MS₂. Other authors used the MSO₁ and MSO₂ notations, and Courcelle came to use it later as well; see, e.g., Courcelle (2018). Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Courcelle, Bruno (2018), "Fly-automata for checking MSO₂ graph properties", Discrete Applied Mathematics, 245: 236–252, arXiv:1511.08605, doi:10.1016/j.dam.2016.10.018, MR 3804787
Fagin, Stockmeyer & Vardi (1995). Fagin, Ronald; Stockmeyer, Larry J.; Vardi, Moshe Y. (1995), "On monadic NP vs monadic co-NP", Information and Computation, 120 (1): 78–92, doi:10.1006/inco.1995.1100, MR 1340807
Courcelle & Engelfriet (2012); Libkin (2004), Corollary 7.24, pp. 126–127. Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Libkin, Leonid (2004), Elements of finite model theory, Texts in Theoretical Computer Science: An EATCS Series, Springer-Verlag, Berlin, doi:10.1007/978-3-662-07003-1, ISBN 3-540-21202-7, MR 2102513, S2CID 30176939
Elberfeld, Jakoby & Tantau (2010). Elberfeld, Michael; Jakoby, Andreas; Tantau, Till (October 2010), "Logspace Versions of the Theorems of Bodlaender and Courcelle" (PDF), Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 143–152, doi:10.1109/FOCS.2010.21, S2CID 1820251
Grohe (2001); Kawarabayashi & Reed (2007). Grohe, Martin (2001), "Computing crossing numbers in quadratic time", Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (STOC '01), pp. 231–236, arXiv:cs/0009010, doi:10.1145/380752.380805, S2CID 724544 Kawarabayashi, Ken-ichi; Reed, Bruce (2007), "Computing crossing number in linear time", Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing (STOC '07), pp. 382–390, doi:10.1145/1250790.1250848, S2CID 13000831
Courcelle & Oum (2007). Courcelle, Bruno; Oum, Sang-il (2007), "Vertex-minors, monadic second-order logic, and a conjecture by Seese" (PDF), Journal of Combinatorial Theory, Series B, 97 (1): 91–126, doi:10.1016/j.jctb.2006.04.003, MR 2278126
Seese (1991). Seese, D. (1991), "The structure of the models of decidable monadic theories of graphs", Annals of Pure and Applied Logic, 53 (2): 169–195, doi:10.1016/0168-0072(91)90054-P, MR 1114848

informatik-rub.de

Zeume (2017). Zeume, Thomas (2017), "The dynamic descriptive complexity of $k$ -clique" (PDF), Information and Computation, 256: 9–22, doi:10.1016/j.ic.2017.04.005, MR 3705411, S2CID 1412001

jstor.org

For instance, Henson (1972) requires directed graphs to be described by an asymmetric relation, meaning that loops and 2-cycles are both disallowed, giving the oriented graphs. Henson, C. Ward (1972), "Countable homogeneous relational structures and $\aleph _{0}$ -categorical theories", The Journal of Symbolic Logic, 37: 494–500, doi:10.2307/2272734, JSTOR 2272734, MR 0321727, S2CID 40662635
Koncewicz (1973). Koncewicz, Leszek (1973), "Definability of classes of graphs in the first order predicate calculus with identity", Polish Academy of Sciences, 32: 159–190, doi:10.1007/BF02123839, JSTOR 20014678, MR 0351796, S2CID 189786935
Glebskiĭ et al. (1969); Fagin (1976) Glebskiĭ, Ju. V.; Kogan, D. I.; Liogon'kiĭ, M. I.; Talanov, V. A. (1969), "Volume and fraction of satisfiability of formulas of the lower predicate calculus", Otdelenie Matematiki, Mekhaniki i Kibernetiki Akademii Nauk Ukrainskoĭ SSR: Kibernetika (2): 17–27, MR 0300882 Fagin, Ronald (1976), "Probabilities on finite models", Journal of Symbolic Logic, 41 (1): 50–58, doi:10.1017/s0022481200051756, JSTOR 2272945, MR 0476480, S2CID 2563318
Shelah & Spencer (1988); Spencer (2001). Shelah, Saharon; Spencer, Joel (1988), "Zero-one laws for sparse random graphs", Journal of the American Mathematical Society, 1 (1): 97–115, doi:10.2307/1990968, JSTOR 1990968, MR 0924703 Spencer, Joel (2001), The Strange Logic of Random Graphs, Algorithms and Combinatorics, vol. 22, Springer-Verlag, Berlin, doi:10.1007/978-3-662-04538-1, ISBN 3-540-41654-4, MR 1847951
Hella, Kolaitis & Luosto (1996). Hella, Lauri; Kolaitis, Phokion G.; Luosto, Kerkko (1996), "Almost everywhere equivalence of logics in finite model theory", The Bulletin of Symbolic Logic, 2 (4): 422–443, doi:10.2307/421173, JSTOR 421173, MR 1460316, S2CID 16411368

kaist.ac.kr

mathsci.kaist.ac.kr

Courcelle & Oum (2007). Courcelle, Bruno; Oum, Sang-il (2007), "Vertex-minors, monadic second-order logic, and a conjecture by Seese" (PDF), Journal of Combinatorial Theory, Series B, 97 (1): 91–126, doi:10.1016/j.jctb.2006.04.003, MR 2278126

labri.fr

Courcelle (1996). Courcelle, Bruno (1996), "On the expression of graph properties in some fragments of monadic second-order logic" (PDF), in Immerman, Neil; Kolaitis, Phokion G. (eds.), Proc. Descr. Complex. Finite Models, DIMACS, vol. 31, Amer. Math. Soc., pp. 33–62, CiteSeerX 10.1.1.55.5184, MR 1451381

psu.edu

citeseerx.ist.psu.edu

Nešetřil & Ossona de Mendez (2012), 18.3 The Subgraph Isomorphism Problem and Boolean Queries, pp. 400–401; Dvořák, Kráľ & Thomas (2010); Grohe, Kreutzer & Siebertz (2014). Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer-Verlag, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058 Dvořák, Zdeněk; Kráľ, Daniel; Thomas, Robin (2010), "Deciding first-order properties for sparse graphs", Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 133–142, CiteSeerX 10.1.1.170.9781, doi:10.1109/FOCS.2010.20, ISBN 978-0-7695-4244-7, MR 3024787, S2CID 15264036 Grohe, Martin; Kreutzer, Stephan; Siebertz, Sebastian (2014), "Deciding first-order properties of nowhere dense graphs", Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC '14), New York: ACM, pp. 89–98, arXiv:1311.3899, doi:10.1145/2591796.2591851, ISBN 978-1-4503-2710-7, S2CID 13297133
Courcelle (1996). Courcelle, Bruno (1996), "On the expression of graph properties in some fragments of monadic second-order logic" (PDF), in Immerman, Neil; Kolaitis, Phokion G. (eds.), Proc. Descr. Complex. Finite Models, DIMACS, vol. 31, Amer. Math. Soc., pp. 33–62, CiteSeerX 10.1.1.55.5184, MR 1451381

semanticscholar.org

api.semanticscholar.org

Verbitsky & Zhukovskii (2019). Verbitsky, Oleg; Zhukovskii, Maksim (2019), "Tight bounds on the asymptotic descriptive complexity of subgraph isomorphism", ACM Transactions on Computational Logic, 20 (2): A9:1–A9:18, arXiv:1802.02143, doi:10.1145/3303881, MR 3942556, S2CID 3603039
Zeume (2017). Zeume, Thomas (2017), "The dynamic descriptive complexity of $k$ -clique" (PDF), Information and Computation, 256: 9–22, doi:10.1016/j.ic.2017.04.005, MR 3705411, S2CID 1412001
Goldberg (1993). Goldberg, Leslie Ann (1993), "Polynomial space polynomial delay algorithms for listing families of graphs", Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing (STOC '93), New York, NY, USA: ACM, pp. 218–225, doi:10.1145/167088.167160, ISBN 0-89791-591-7, S2CID 6305108
For instance, Henson (1972) requires directed graphs to be described by an asymmetric relation, meaning that loops and 2-cycles are both disallowed, giving the oriented graphs. Henson, C. Ward (1972), "Countable homogeneous relational structures and $\aleph _{0}$ -categorical theories", The Journal of Symbolic Logic, 37: 494–500, doi:10.2307/2272734, JSTOR 2272734, MR 0321727, S2CID 40662635
Koncewicz (1973). Koncewicz, Leszek (1973), "Definability of classes of graphs in the first order predicate calculus with identity", Polish Academy of Sciences, 32: 159–190, doi:10.1007/BF02123839, JSTOR 20014678, MR 0351796, S2CID 189786935
Glebskiĭ et al. (1969); Fagin (1976) Glebskiĭ, Ju. V.; Kogan, D. I.; Liogon'kiĭ, M. I.; Talanov, V. A. (1969), "Volume and fraction of satisfiability of formulas of the lower predicate calculus", Otdelenie Matematiki, Mekhaniki i Kibernetiki Akademii Nauk Ukrainskoĭ SSR: Kibernetika (2): 17–27, MR 0300882 Fagin, Ronald (1976), "Probabilities on finite models", Journal of Symbolic Logic, 41 (1): 50–58, doi:10.1017/s0022481200051756, JSTOR 2272945, MR 0476480, S2CID 2563318
Spencer (1990). Spencer, Joel (1990), "Infinite spectra in the first order theory of graphs", Combinatorica, 10 (1): 95–102, doi:10.1007/BF02122699, MR 1075070, S2CID 27770505
Nešetřil & Ossona de Mendez (2012), 18.3 The Subgraph Isomorphism Problem and Boolean Queries, pp. 400–401; Dvořák, Kráľ & Thomas (2010); Grohe, Kreutzer & Siebertz (2014). Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer-Verlag, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058 Dvořák, Zdeněk; Kráľ, Daniel; Thomas, Robin (2010), "Deciding first-order properties for sparse graphs", Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 133–142, CiteSeerX 10.1.1.170.9781, doi:10.1109/FOCS.2010.20, ISBN 978-0-7695-4244-7, MR 3024787, S2CID 15264036 Grohe, Martin; Kreutzer, Stephan; Siebertz, Sebastian (2014), "Deciding first-order properties of nowhere dense graphs", Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC '14), New York: ACM, pp. 89–98, arXiv:1311.3899, doi:10.1145/2591796.2591851, ISBN 978-1-4503-2710-7, S2CID 13297133
Pikhurko, Spencer & Verbitsky (2006). Pikhurko, Oleg; Spencer, Joel; Verbitsky, Oleg (2006), "Succinct definitions in the first order theory of graphs", Annals of Pure and Applied Logic, 139 (1–3): 74–109, arXiv:math/0401307, doi:10.1016/j.apal.2005.04.003, MR 2206252, S2CID 3041191
Verbitsky (2005). Verbitsky, Oleg (2005), "The first order definability of graphs with separators via the Ehrenfeucht game", Theoretical Computer Science, 343 (1–2): 158–176, arXiv:math/0401361, doi:10.1016/j.tcs.2005.05.003, MR 2168849, S2CID 17886484
Ebbinghaus & Flum (1995). Parys (2014) writes that this undecidability result is well known, and attributes it to Trahtenbrot (1950) on the undecidability of first-order satisfiability for more general classes of finite structures. Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995), Finite Model Theory, Springer Monographs in Mathematics (2nd ed.), Springer, p. 129, doi:10.1007/3-540-28788-4 Parys, Paweł (2014), "First-order logic on CPDA graphs", Computer science—theory and applications, Lecture Notes in Computer Science, vol. 8476, New York: Springer-Verlag, pp. 300–313, doi:10.1007/978-3-319-06686-8_23, MR 3218557, S2CID 31640587 Trahtenbrot, B. A. (1950), "The impossibility of an algorithm for the decision problem for finite domains", Doklady Akademii Nauk SSSR, New Series, 70: 569–572, MR 0033784
Hella, Kolaitis & Luosto (1996). Hella, Lauri; Kolaitis, Phokion G.; Luosto, Kerkko (1996), "Almost everywhere equivalence of logics in finite model theory", The Bulletin of Symbolic Logic, 2 (4): 422–443, doi:10.2307/421173, JSTOR 421173, MR 1460316, S2CID 16411368
Laubner (2010). Laubner, Bastian (2010), "Capturing polynomial time on interval graphs", 25th Annual IEEE Symposium on Logic in Computer Science (LICS 2010), Los Alamitos, California: IEEE Computer Society, pp. 199–208, arXiv:0911.3799, doi:10.1109/LICS.2010.42, MR 2963094, S2CID 1450409
Courcelle & Engelfriet (2012); Libkin (2004), Corollary 7.24, pp. 126–127. Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Libkin, Leonid (2004), Elements of finite model theory, Texts in Theoretical Computer Science: An EATCS Series, Springer-Verlag, Berlin, doi:10.1007/978-3-662-07003-1, ISBN 3-540-21202-7, MR 2102513, S2CID 30176939
Elberfeld, Jakoby & Tantau (2010). Elberfeld, Michael; Jakoby, Andreas; Tantau, Till (October 2010), "Logspace Versions of the Theorems of Bodlaender and Courcelle" (PDF), Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 143–152, doi:10.1109/FOCS.2010.21, S2CID 1820251
Grohe (2001); Kawarabayashi & Reed (2007). Grohe, Martin (2001), "Computing crossing numbers in quadratic time", Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (STOC '01), pp. 231–236, arXiv:cs/0009010, doi:10.1145/380752.380805, S2CID 724544 Kawarabayashi, Ken-ichi; Reed, Bruce (2007), "Computing crossing number in linear time", Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing (STOC '07), pp. 382–390, doi:10.1145/1250790.1250848, S2CID 13000831

tum.de

wwwmayr.in.tum.de

Elberfeld, Jakoby & Tantau (2010). Elberfeld, Michael; Jakoby, Andreas; Tantau, Till (October 2010), "Logspace Versions of the Theorems of Bodlaender and Courcelle" (PDF), Proc. 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 143–152, doi:10.1109/FOCS.2010.21, S2CID 1820251

zbmath.org

These definitions can be found e.g. in Courcelle & Engelfriet (2012), p. 69, under the slightly different notation MS₁ and MS₂. Other authors used the MSO₁ and MSO₂ notations, and Courcelle came to use it later as well; see, e.g., Courcelle (2018). Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Courcelle, Bruno (2018), "Fly-automata for checking MSO₂ graph properties", Discrete Applied Mathematics, 245: 236–252, arXiv:1511.08605, doi:10.1016/j.dam.2016.10.018, MR 3804787
Courcelle & Engelfriet (2012); Libkin (2004), Corollary 7.24, pp. 126–127. Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006 Libkin, Leonid (2004), Elements of finite model theory, Texts in Theoretical Computer Science: An EATCS Series, Springer-Verlag, Berlin, doi:10.1007/978-3-662-07003-1, ISBN 3-540-21202-7, MR 2102513, S2CID 30176939
Courcelle & Engelfriet (2012). Courcelle, Bruno; Engelfriet, Joost (2012), Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138, Cambridge University Press, ISBN 9781139644006, Zbl 1257.68006