Apollonian network (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Apollonian network" in English language version.

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17doi.org

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9arxiv.org

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7semanticscholar.org

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ams.org (Global: 451^st place; English: 277^th place)

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Nishizeki (1980). Nishizeki, Takao (1980), "A 1-tough non-Hamiltonian maximal planar graph", Discrete Mathematics, 30 (3): 305–307, doi:10.1016/0012-365X(80)90240-X, MR 0573648.
The equivalence of planar 3-trees and chordal maximal planar graphs was stated without proof by Patil (1986). For a proof, see Markenzon, Justel & Paciornik (2006). For a more general characterization of chordal planar graphs, and an efficient recognition algorithm for these graphs, see Kumar & Madhavan (1989). The observation that every chordal polyhedral graph is maximal planar was stated explicitly by Gerlach (2004). Patil, H. P. (1986), "On the structure of k-trees", Journal of Combinatorics, Information and System Sciences, 11 (2–4): 57–64, MR 0966069. Markenzon, L.; Justel, C. M.; Paciornik, N. (2006), "Subclasses of k-trees: Characterization and recognition", Discrete Applied Mathematics, 154 (5): 818–825, doi:10.1016/j.dam.2005.05.021, MR 2207565. Kumar, P. Sreenivasa; Madhavan, C. E. Veni (1989), "A new class of separators and planarity of chordal graphs", Foundations of Software Technology and Theoretical Computer Science, Ninth Conference, Bangalore, India December 19–21, 1989, Proceedings, Lecture Notes in Computer Science, vol. 405, Springer-Verlag, pp. 30–43, doi:10.1007/3-540-52048-1_30, ISBN 978-3-540-52048-1, MR 1048636. Gerlach, T. (2004), "Toughness and Hamiltonicity of a class of planar graphs", Discrete Mathematics, 286 (1–2): 61–65, doi:10.1016/j.disc.2003.11.046, MR 2084280.
Felsner & Zickfeld (2008); Bernardi & Bonichon (2009). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657. Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248.
The two forbidden minors for planar graphs are given by Wagner's theorem. For the forbidden minors for partial 3-trees (which include also the nonplanar Wagner graph) see Arnborg, Proskurowski & Corniel (1986) and Bodlaender (1998). For direct proofs that the octahedral graph and the pentagonal-prism graph are the only two planar forbidden minors, see Dai & Sato (1990) and El-Mallah & Colbourn (1990). Arnborg, S.; Proskurowski, A.; Corniel, D. (1986), Forbidden Minors Characterization of Partial 3-trees, Technical Report CIS-TR-86-07, Dept. of Computer and Information Science, University of Oregon. As cited by El-Mallah & Colbourn (1990). Bodlaender, Hans L. (1998), "A partial k-arboretum of graphs with bounded treewidth", Theoretical Computer Science, 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4, hdl:1874/18312, MR 1647486. Dai, Wayne Wei-Ming; Sato, Masao (1990), "Minimal forbidden minor characterization of planar partial 3-trees and application to circuit layout", IEEE International Symposium on Circuits and Systems, vol. 4, pp. 2677–2681, doi:10.1109/ISCAS.1990.112560, S2CID 122926229 El-Mallah, Ehab S.; Colbourn, Charles J. (1990), "On two dual classes of planar graphs", Discrete Mathematics, 80 (1): 21–40, doi:10.1016/0012-365X(90)90293-Q, MR 1045921.
Politof (1983) introduced the Δ-Y reducible planar graphs and characterized them in terms of forbidden homeomorphic subgraphs. The duality between the Δ-Y and Y-Δ reducible graphs, the forbidden minor characterizations of both classes, and the connection to planar partial 3-trees are all from El-Mallah & Colbourn (1990). Politof, T. (1983), A characterization and efficient reliability computation of Δ-Y reducible networks, Ph.D. thesis, University of California, Berkeley. As cited by El-Mallah & Colbourn (1990). El-Mallah, Ehab S.; Colbourn, Charles J. (1990), "On two dual classes of planar graphs", Discrete Mathematics, 80 (1): 21–40, doi:10.1016/0012-365X(90)90293-Q, MR 1045921.
For the characterization in terms of the maximum number of triangles in a planar graph, see Hakimi & Schmeichel (1979). Alon & Caro (1984) quote this result and provide the characterizations in terms of the isomorphism classes of blocks and numbers of blocks. The bound on the total number of cliques follows easily from the bounds on triangles and K₄ subgraphs, and is also stated explicitly by Wood (2007), who provides an Apollonian network as an example showing that this bound is tight. For generalizations of these bounds to nonplanar surfaces, see Dujmović et al. (2009). Hakimi, S. L.; Schmeichel, E. F. (1979), "On the number of cycles of length k in a maximal planar graph", Journal of Graph Theory, 3 (1): 69–86, doi:10.1002/jgt.3190030108, MR 0519175. Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Wood, David R. (2007), "On the maximum number of cliques in a graph", Graphs and Combinatorics, 23 (3): 337–352, arXiv:math/0602191, doi:10.1007/s00373-007-0738-8, MR 2320588, S2CID 46700417. Dujmović, Vida; Fijavž, Gašper; Joret, Gwenaël; Wood, David R. (2009), "The maximum number of cliques in a graph embedded in a surface", European Journal of Combinatorics, 32 (8): 1244–1252, arXiv:0906.4142, Bibcode:2009arXiv0906.4142D, doi:10.1016/j.ejc.2011.04.001, S2CID 1733300.
For subdividing a triangle with rational side lengths so that the smaller triangles also have rational side lengths, see Almering (1963). For progress on the general problem of finding planar drawings with rational edge lengths, see Geelen, Guo & McKinnon (2008). Almering, J. H. J. (1963), "Rational quadrilaterals", Indagationes Mathematicae, 25: 192–199, doi:10.1016/S1385-7258(63)50020-1, MR 0147447. Geelen, Jim; Guo, Anjie; McKinnon, David (2008), "Straight line embeddings of cubic planar graphs with integer edge lengths", Journal of Graph Theory, 58 (3): 270–274, doi:10.1002/jgt.20304, MR 2419522.
Plummer (1992). Plummer, Michael D. (1992), "Extending matchings in planar graphs IV", Discrete Mathematics, 109 (1–3): 207–219, doi:10.1016/0012-365X(92)90292-N, MR 1192384.
See Nishizeki (1980) for a 1-tough non-Hamiltonian example, Böhme, Harant & Tkáč (1999) for the proof that Apollonian networks with greater toughness are Hamiltonian, and Gerlach (2004) for an extension of this result to a wider class of planar graphs. Nishizeki, Takao (1980), "A 1-tough non-Hamiltonian maximal planar graph", Discrete Mathematics, 30 (3): 305–307, doi:10.1016/0012-365X(80)90240-X, MR 0573648. Böhme, Thomas; Harant, Jochen; Tkáč, Michal (1999), "More than one tough chordal planar graphs are Hamiltonian", Journal of Graph Theory, 32 (4): 405–410, doi:10.1002/(SICI)1097-0118(199912)32:4<405::AID-JGT8>3.3.CO;2-Q, MR 1722793. Gerlach, T. (2004), "Toughness and Hamiltonicity of a class of planar graphs", Discrete Mathematics, 286 (1–2): 61–65, doi:10.1016/j.disc.2003.11.046, MR 2084280.
Grünbaum (1963); Grünbaum (1967). Grünbaum, Branko (1963), "Unambiguous polyhedral graphs", Israel Journal of Mathematics, 1 (4): 235–238, doi:10.1007/BF02759726, MR 0185506, S2CID 121075042. Grünbaum, Branko (1967), Convex Polytopes, Wiley Interscience.
Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008). Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Zickfeld, Florian; Ziegler, Günter M. (2006), "Integer realizations of stacked polytopes", Workshop on Geometric and Topological Combinatorics (PDF). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657.
Albenque & Marckert (2008); Bernardi & Bonichon (2009); Jiménez & Kiwi (2010). Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248. Jiménez, Andrea; Kiwi, Marcos (2010), Counting perfect matchings of cubic graphs in the geometric dual, arXiv:1010.5918, Bibcode:2010arXiv1010.5918J.

arxiv.org (Global: 69^th place; English: 59^th place)

Felsner & Zickfeld (2008); Bernardi & Bonichon (2009). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657. Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248.
For the characterization in terms of the maximum number of triangles in a planar graph, see Hakimi & Schmeichel (1979). Alon & Caro (1984) quote this result and provide the characterizations in terms of the isomorphism classes of blocks and numbers of blocks. The bound on the total number of cliques follows easily from the bounds on triangles and K₄ subgraphs, and is also stated explicitly by Wood (2007), who provides an Apollonian network as an example showing that this bound is tight. For generalizations of these bounds to nonplanar surfaces, see Dujmović et al. (2009). Hakimi, S. L.; Schmeichel, E. F. (1979), "On the number of cycles of length k in a maximal planar graph", Journal of Graph Theory, 3 (1): 69–86, doi:10.1002/jgt.3190030108, MR 0519175. Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Wood, David R. (2007), "On the maximum number of cliques in a graph", Graphs and Combinatorics, 23 (3): 337–352, arXiv:math/0602191, doi:10.1007/s00373-007-0738-8, MR 2320588, S2CID 46700417. Dujmović, Vida; Fijavž, Gašper; Joret, Gwenaël; Wood, David R. (2009), "The maximum number of cliques in a graph embedded in a surface", European Journal of Combinatorics, 32 (8): 1244–1252, arXiv:0906.4142, Bibcode:2009arXiv0906.4142D, doi:10.1016/j.ejc.2011.04.001, S2CID 1733300.
Jiménez & Kiwi (2010). Jiménez, Andrea; Kiwi, Marcos (2010), Counting perfect matchings of cubic graphs in the geometric dual, arXiv:1010.5918, Bibcode:2010arXiv1010.5918J.
Tsourakakis (2011) Tsourakakis, Charalampos E. (2011), The Degree Sequence of Random Apollonian Networks, arXiv:1106.1940.
Zhou et al. (2004); Zhou, Yan & Wang (2005). Zhou, Tao; Yan, Gang; Zhou, Pei-Ling; Fu, Zhong-Qian; Wang, Bing-Hong (2004), Random Apollonian networks, arXiv:cond-mat/0409414v2, Bibcode:2004cond.mat..9414Z. Zhou, Tao; Yan, Gang; Wang, Bing-Hong (2005), "Maximal planar networks with large clustering coefficient and power-law degree distribution", Physical Review E, 71 (4) 046141, arXiv:cond-mat/0412448, Bibcode:2005PhRvE..71d6141Z, doi:10.1103/PhysRevE.71.046141, PMID 15903760, S2CID 21740312.
Frieze & Tsourakakis (2011) Frieze, Alan M.; Tsourakakis, Charalampos E. (2011), High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks, arXiv:1104.5259, Bibcode:2011arXiv1104.5259F.
Albenque & Marckert (2008);Zhang et al. (2008). Zhang, Zhongzhi; Chen, Lichao; Zhou, Shuigeng; Fang, Lujun; Guan, Jihong; Zou, Tao (2008), "Analytical solution of average path length for Apollonian networks", Physical Review E, 77 (1) 017102, arXiv:0706.3491, Bibcode:2008PhRvE..77a7102Z, doi:10.1103/PhysRevE.77.017102, PMID 18351964, S2CID 30404208.
Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008). Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Zickfeld, Florian; Ziegler, Günter M. (2006), "Integer realizations of stacked polytopes", Workshop on Geometric and Topological Combinatorics (PDF). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657.
Albenque & Marckert (2008); Bernardi & Bonichon (2009); Jiménez & Kiwi (2010). Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248. Jiménez, Andrea; Kiwi, Marcos (2010), Counting perfect matchings of cubic graphs in the geometric dual, arXiv:1010.5918, Bibcode:2010arXiv1010.5918J.

cccg.ca (Global: low place; English: low place)

For the drawings with integer coordinates, see Mondal et al. (2010), and for the drawings on a given vertex set, see Nishat, Mondal & Rahman (2011). Mondal, Debajyoti; Nishat, Rahnuma Islam; Rahman, Md. Saidur; Alam, Muhammad Jawaherul (2010), "Minimum-area drawings of plane 3-trees", Canadian Conference on Computational Geometry (PDF). Nishat, Rahnuma Islam; Mondal, Debajyoti; Rahman, Md. Saidur (2011), "Point-set embeddings of plane 3-trees", Graph Drawing, 18th International Symposium, GD 2010, Konstanz, Germany, September 21–24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 6502, Springer-Verlag, pp. 317–328, doi:10.1007/978-3-642-18469-7_29, ISBN 978-3-642-18468-0.

doi.org (Global: 2^nd place; English: 2^nd place)

Nishizeki (1980). Nishizeki, Takao (1980), "A 1-tough non-Hamiltonian maximal planar graph", Discrete Mathematics, 30 (3): 305–307, doi:10.1016/0012-365X(80)90240-X, MR 0573648.
The equivalence of planar 3-trees and chordal maximal planar graphs was stated without proof by Patil (1986). For a proof, see Markenzon, Justel & Paciornik (2006). For a more general characterization of chordal planar graphs, and an efficient recognition algorithm for these graphs, see Kumar & Madhavan (1989). The observation that every chordal polyhedral graph is maximal planar was stated explicitly by Gerlach (2004). Patil, H. P. (1986), "On the structure of k-trees", Journal of Combinatorics, Information and System Sciences, 11 (2–4): 57–64, MR 0966069. Markenzon, L.; Justel, C. M.; Paciornik, N. (2006), "Subclasses of k-trees: Characterization and recognition", Discrete Applied Mathematics, 154 (5): 818–825, doi:10.1016/j.dam.2005.05.021, MR 2207565. Kumar, P. Sreenivasa; Madhavan, C. E. Veni (1989), "A new class of separators and planarity of chordal graphs", Foundations of Software Technology and Theoretical Computer Science, Ninth Conference, Bangalore, India December 19–21, 1989, Proceedings, Lecture Notes in Computer Science, vol. 405, Springer-Verlag, pp. 30–43, doi:10.1007/3-540-52048-1_30, ISBN 978-3-540-52048-1, MR 1048636. Gerlach, T. (2004), "Toughness and Hamiltonicity of a class of planar graphs", Discrete Mathematics, 286 (1–2): 61–65, doi:10.1016/j.disc.2003.11.046, MR 2084280.
The fact that Apollonian networks minimize the number of colorings with more than four of colors was shown in a dual form for colorings of maps by Birkhoff (1930). Birkhoff, George D. (1930), "On the number of ways of colouring a map", Proceedings of the Edinburgh Mathematical Society, (2), 2 (2): 83–91, doi:10.1017/S0013091500007598.
Felsner & Zickfeld (2008); Bernardi & Bonichon (2009). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657. Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248.
The two forbidden minors for planar graphs are given by Wagner's theorem. For the forbidden minors for partial 3-trees (which include also the nonplanar Wagner graph) see Arnborg, Proskurowski & Corniel (1986) and Bodlaender (1998). For direct proofs that the octahedral graph and the pentagonal-prism graph are the only two planar forbidden minors, see Dai & Sato (1990) and El-Mallah & Colbourn (1990). Arnborg, S.; Proskurowski, A.; Corniel, D. (1986), Forbidden Minors Characterization of Partial 3-trees, Technical Report CIS-TR-86-07, Dept. of Computer and Information Science, University of Oregon. As cited by El-Mallah & Colbourn (1990). Bodlaender, Hans L. (1998), "A partial k-arboretum of graphs with bounded treewidth", Theoretical Computer Science, 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4, hdl:1874/18312, MR 1647486. Dai, Wayne Wei-Ming; Sato, Masao (1990), "Minimal forbidden minor characterization of planar partial 3-trees and application to circuit layout", IEEE International Symposium on Circuits and Systems, vol. 4, pp. 2677–2681, doi:10.1109/ISCAS.1990.112560, S2CID 122926229 El-Mallah, Ehab S.; Colbourn, Charles J. (1990), "On two dual classes of planar graphs", Discrete Mathematics, 80 (1): 21–40, doi:10.1016/0012-365X(90)90293-Q, MR 1045921.
Politof (1983) introduced the Δ-Y reducible planar graphs and characterized them in terms of forbidden homeomorphic subgraphs. The duality between the Δ-Y and Y-Δ reducible graphs, the forbidden minor characterizations of both classes, and the connection to planar partial 3-trees are all from El-Mallah & Colbourn (1990). Politof, T. (1983), A characterization and efficient reliability computation of Δ-Y reducible networks, Ph.D. thesis, University of California, Berkeley. As cited by El-Mallah & Colbourn (1990). El-Mallah, Ehab S.; Colbourn, Charles J. (1990), "On two dual classes of planar graphs", Discrete Mathematics, 80 (1): 21–40, doi:10.1016/0012-365X(90)90293-Q, MR 1045921.
For the characterization in terms of the maximum number of triangles in a planar graph, see Hakimi & Schmeichel (1979). Alon & Caro (1984) quote this result and provide the characterizations in terms of the isomorphism classes of blocks and numbers of blocks. The bound on the total number of cliques follows easily from the bounds on triangles and K₄ subgraphs, and is also stated explicitly by Wood (2007), who provides an Apollonian network as an example showing that this bound is tight. For generalizations of these bounds to nonplanar surfaces, see Dujmović et al. (2009). Hakimi, S. L.; Schmeichel, E. F. (1979), "On the number of cycles of length k in a maximal planar graph", Journal of Graph Theory, 3 (1): 69–86, doi:10.1002/jgt.3190030108, MR 0519175. Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Wood, David R. (2007), "On the maximum number of cliques in a graph", Graphs and Combinatorics, 23 (3): 337–352, arXiv:math/0602191, doi:10.1007/s00373-007-0738-8, MR 2320588, S2CID 46700417. Dujmović, Vida; Fijavž, Gašper; Joret, Gwenaël; Wood, David R. (2009), "The maximum number of cliques in a graph embedded in a surface", European Journal of Combinatorics, 32 (8): 1244–1252, arXiv:0906.4142, Bibcode:2009arXiv0906.4142D, doi:10.1016/j.ejc.2011.04.001, S2CID 1733300.
Elcock, Gargantini & Walsh (1987). Elcock, E. W.; Gargantini, I.; Walsh, T. R. (1987), "Triangular decomposition", Image and Vision Computing, 5 (3): 225–231, doi:10.1016/0262-8856(87)90053-9.
For subdividing a triangle with rational side lengths so that the smaller triangles also have rational side lengths, see Almering (1963). For progress on the general problem of finding planar drawings with rational edge lengths, see Geelen, Guo & McKinnon (2008). Almering, J. H. J. (1963), "Rational quadrilaterals", Indagationes Mathematicae, 25: 192–199, doi:10.1016/S1385-7258(63)50020-1, MR 0147447. Geelen, Jim; Guo, Anjie; McKinnon, David (2008), "Straight line embeddings of cubic planar graphs with integer edge lengths", Journal of Graph Theory, 58 (3): 270–274, doi:10.1002/jgt.20304, MR 2419522.
For the drawings with integer coordinates, see Mondal et al. (2010), and for the drawings on a given vertex set, see Nishat, Mondal & Rahman (2011). Mondal, Debajyoti; Nishat, Rahnuma Islam; Rahman, Md. Saidur; Alam, Muhammad Jawaherul (2010), "Minimum-area drawings of plane 3-trees", Canadian Conference on Computational Geometry (PDF). Nishat, Rahnuma Islam; Mondal, Debajyoti; Rahman, Md. Saidur (2011), "Point-set embeddings of plane 3-trees", Graph Drawing, 18th International Symposium, GD 2010, Konstanz, Germany, September 21–24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 6502, Springer-Verlag, pp. 317–328, doi:10.1007/978-3-642-18469-7_29, ISBN 978-3-642-18468-0.
Plummer (1992). Plummer, Michael D. (1992), "Extending matchings in planar graphs IV", Discrete Mathematics, 109 (1–3): 207–219, doi:10.1016/0012-365X(92)90292-N, MR 1192384.
Zhou et al. (2004); Zhou, Yan & Wang (2005). Zhou, Tao; Yan, Gang; Zhou, Pei-Ling; Fu, Zhong-Qian; Wang, Bing-Hong (2004), Random Apollonian networks, arXiv:cond-mat/0409414v2, Bibcode:2004cond.mat..9414Z. Zhou, Tao; Yan, Gang; Wang, Bing-Hong (2005), "Maximal planar networks with large clustering coefficient and power-law degree distribution", Physical Review E, 71 (4) 046141, arXiv:cond-mat/0412448, Bibcode:2005PhRvE..71d6141Z, doi:10.1103/PhysRevE.71.046141, PMID 15903760, S2CID 21740312.
Albenque & Marckert (2008);Zhang et al. (2008). Zhang, Zhongzhi; Chen, Lichao; Zhou, Shuigeng; Fang, Lujun; Guan, Jihong; Zou, Tao (2008), "Analytical solution of average path length for Apollonian networks", Physical Review E, 77 (1) 017102, arXiv:0706.3491, Bibcode:2008PhRvE..77a7102Z, doi:10.1103/PhysRevE.77.017102, PMID 18351964, S2CID 30404208.
See Nishizeki (1980) for a 1-tough non-Hamiltonian example, Böhme, Harant & Tkáč (1999) for the proof that Apollonian networks with greater toughness are Hamiltonian, and Gerlach (2004) for an extension of this result to a wider class of planar graphs. Nishizeki, Takao (1980), "A 1-tough non-Hamiltonian maximal planar graph", Discrete Mathematics, 30 (3): 305–307, doi:10.1016/0012-365X(80)90240-X, MR 0573648. Böhme, Thomas; Harant, Jochen; Tkáč, Michal (1999), "More than one tough chordal planar graphs are Hamiltonian", Journal of Graph Theory, 32 (4): 405–410, doi:10.1002/(SICI)1097-0118(199912)32:4<405::AID-JGT8>3.3.CO;2-Q, MR 1722793. Gerlach, T. (2004), "Toughness and Hamiltonicity of a class of planar graphs", Discrete Mathematics, 286 (1–2): 61–65, doi:10.1016/j.disc.2003.11.046, MR 2084280.
Grünbaum (1963); Grünbaum (1967). Grünbaum, Branko (1963), "Unambiguous polyhedral graphs", Israel Journal of Mathematics, 1 (4): 235–238, doi:10.1007/BF02759726, MR 0185506, S2CID 121075042. Grünbaum, Branko (1967), Convex Polytopes, Wiley Interscience.
Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008). Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Zickfeld, Florian; Ziegler, Günter M. (2006), "Integer realizations of stacked polytopes", Workshop on Geometric and Topological Combinatorics (PDF). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657.
Albenque & Marckert (2008); Bernardi & Bonichon (2009); Jiménez & Kiwi (2010). Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248. Jiménez, Andrea; Kiwi, Marcos (2010), Counting perfect matchings of cubic graphs in the geometric dual, arXiv:1010.5918, Bibcode:2010arXiv1010.5918J.

emis.de (Global: low place; English: low place)

Felsner & Zickfeld (2008); Bernardi & Bonichon (2009). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657. Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248.
Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008). Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Zickfeld, Florian; Ziegler, Günter M. (2006), "Integer realizations of stacked polytopes", Workshop on Geometric and Topological Combinatorics (PDF). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657.

gatech.edu (Global: 4,448^th place; English: 2,923^rd place)

people.math.gatech.edu

Fowler (1998). Fowler, Thomas (1998), Unique Coloring of Planar Graphs (PDF), Ph.D. thesis, Georgia Institute of Technology Mathematics Department.

handle.net (Global: 102^nd place; English: 76^th place)

hdl.handle.net

The two forbidden minors for planar graphs are given by Wagner's theorem. For the forbidden minors for partial 3-trees (which include also the nonplanar Wagner graph) see Arnborg, Proskurowski & Corniel (1986) and Bodlaender (1998). For direct proofs that the octahedral graph and the pentagonal-prism graph are the only two planar forbidden minors, see Dai & Sato (1990) and El-Mallah & Colbourn (1990). Arnborg, S.; Proskurowski, A.; Corniel, D. (1986), Forbidden Minors Characterization of Partial 3-trees, Technical Report CIS-TR-86-07, Dept. of Computer and Information Science, University of Oregon. As cited by El-Mallah & Colbourn (1990). Bodlaender, Hans L. (1998), "A partial k-arboretum of graphs with bounded treewidth", Theoretical Computer Science, 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4, hdl:1874/18312, MR 1647486. Dai, Wayne Wei-Ming; Sato, Masao (1990), "Minimal forbidden minor characterization of planar partial 3-trees and application to circuit layout", IEEE International Symposium on Circuits and Systems, vol. 4, pp. 2677–2681, doi:10.1109/ISCAS.1990.112560, S2CID 122926229 El-Mallah, Ehab S.; Colbourn, Charles J. (1990), "On two dual classes of planar graphs", Discrete Mathematics, 80 (1): 21–40, doi:10.1016/0012-365X(90)90293-Q, MR 1045921.

harvard.edu (Global: 18^th place; English: 17^th place)

ui.adsabs.harvard.edu

Felsner & Zickfeld (2008); Bernardi & Bonichon (2009). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657. Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248.
For the characterization in terms of the maximum number of triangles in a planar graph, see Hakimi & Schmeichel (1979). Alon & Caro (1984) quote this result and provide the characterizations in terms of the isomorphism classes of blocks and numbers of blocks. The bound on the total number of cliques follows easily from the bounds on triangles and K₄ subgraphs, and is also stated explicitly by Wood (2007), who provides an Apollonian network as an example showing that this bound is tight. For generalizations of these bounds to nonplanar surfaces, see Dujmović et al. (2009). Hakimi, S. L.; Schmeichel, E. F. (1979), "On the number of cycles of length k in a maximal planar graph", Journal of Graph Theory, 3 (1): 69–86, doi:10.1002/jgt.3190030108, MR 0519175. Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Wood, David R. (2007), "On the maximum number of cliques in a graph", Graphs and Combinatorics, 23 (3): 337–352, arXiv:math/0602191, doi:10.1007/s00373-007-0738-8, MR 2320588, S2CID 46700417. Dujmović, Vida; Fijavž, Gašper; Joret, Gwenaël; Wood, David R. (2009), "The maximum number of cliques in a graph embedded in a surface", European Journal of Combinatorics, 32 (8): 1244–1252, arXiv:0906.4142, Bibcode:2009arXiv0906.4142D, doi:10.1016/j.ejc.2011.04.001, S2CID 1733300.
Jiménez & Kiwi (2010). Jiménez, Andrea; Kiwi, Marcos (2010), Counting perfect matchings of cubic graphs in the geometric dual, arXiv:1010.5918, Bibcode:2010arXiv1010.5918J.
Zhou et al. (2004); Zhou, Yan & Wang (2005). Zhou, Tao; Yan, Gang; Zhou, Pei-Ling; Fu, Zhong-Qian; Wang, Bing-Hong (2004), Random Apollonian networks, arXiv:cond-mat/0409414v2, Bibcode:2004cond.mat..9414Z. Zhou, Tao; Yan, Gang; Wang, Bing-Hong (2005), "Maximal planar networks with large clustering coefficient and power-law degree distribution", Physical Review E, 71 (4) 046141, arXiv:cond-mat/0412448, Bibcode:2005PhRvE..71d6141Z, doi:10.1103/PhysRevE.71.046141, PMID 15903760, S2CID 21740312.
Frieze & Tsourakakis (2011) Frieze, Alan M.; Tsourakakis, Charalampos E. (2011), High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks, arXiv:1104.5259, Bibcode:2011arXiv1104.5259F.
Albenque & Marckert (2008);Zhang et al. (2008). Zhang, Zhongzhi; Chen, Lichao; Zhou, Shuigeng; Fang, Lujun; Guan, Jihong; Zou, Tao (2008), "Analytical solution of average path length for Apollonian networks", Physical Review E, 77 (1) 017102, arXiv:0706.3491, Bibcode:2008PhRvE..77a7102Z, doi:10.1103/PhysRevE.77.017102, PMID 18351964, S2CID 30404208.
Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008). Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Zickfeld, Florian; Ziegler, Günter M. (2006), "Integer realizations of stacked polytopes", Workshop on Geometric and Topological Combinatorics (PDF). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657.
Albenque & Marckert (2008); Bernardi & Bonichon (2009); Jiménez & Kiwi (2010). Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248. Jiménez, Andrea; Kiwi, Marcos (2010), Counting perfect matchings of cubic graphs in the geometric dual, arXiv:1010.5918, Bibcode:2010arXiv1010.5918J.

msri.org (Global: low place; English: low place)

Thurston (1978–1981). Thurston, William (1978–1981), The geometry and topology of 3-manifolds, Princeton lecture notes.

nih.gov (Global: 4^th place; English: 4^th place)

pubmed.ncbi.nlm.nih.gov

Zhou et al. (2004); Zhou, Yan & Wang (2005). Zhou, Tao; Yan, Gang; Zhou, Pei-Ling; Fu, Zhong-Qian; Wang, Bing-Hong (2004), Random Apollonian networks, arXiv:cond-mat/0409414v2, Bibcode:2004cond.mat..9414Z. Zhou, Tao; Yan, Gang; Wang, Bing-Hong (2005), "Maximal planar networks with large clustering coefficient and power-law degree distribution", Physical Review E, 71 (4) 046141, arXiv:cond-mat/0412448, Bibcode:2005PhRvE..71d6141Z, doi:10.1103/PhysRevE.71.046141, PMID 15903760, S2CID 21740312.
Albenque & Marckert (2008);Zhang et al. (2008). Zhang, Zhongzhi; Chen, Lichao; Zhou, Shuigeng; Fang, Lujun; Guan, Jihong; Zou, Tao (2008), "Analytical solution of average path length for Apollonian networks", Physical Review E, 77 (1) 017102, arXiv:0706.3491, Bibcode:2008PhRvE..77a7102Z, doi:10.1103/PhysRevE.77.017102, PMID 18351964, S2CID 30404208.

nmsu.edu (Global: 6,873^rd place; English: 4,881^st place)

cs.nmsu.edu

This graph is named after the work of Goldner & Harary (1975); however, it appears earlier in the literature, for instance in Grünbaum (1967), p. 357. Goldner, A.; Harary, F. (1975), "Note on a smallest nonhamiltonian maximal planar graph", Bull. Malaysian Math. Soc., 6 (1): 41–42. See also the same journal 6(2):33 (1975) and 8:104-106 (1977). Reference from listing of Harary's publications. Grünbaum, Branko (1967), Convex Polytopes, Wiley Interscience.

semanticscholar.org (Global: 11^th place; English: 8^th place)

api.semanticscholar.org

Felsner & Zickfeld (2008); Bernardi & Bonichon (2009). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657. Bernardi, Olivier; Bonichon, Nicolas (2009), "Intervals in Catalan lattices and realizers of triangulations", Journal of Combinatorial Theory, Series A, 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248.
The two forbidden minors for planar graphs are given by Wagner's theorem. For the forbidden minors for partial 3-trees (which include also the nonplanar Wagner graph) see Arnborg, Proskurowski & Corniel (1986) and Bodlaender (1998). For direct proofs that the octahedral graph and the pentagonal-prism graph are the only two planar forbidden minors, see Dai & Sato (1990) and El-Mallah & Colbourn (1990). Arnborg, S.; Proskurowski, A.; Corniel, D. (1986), Forbidden Minors Characterization of Partial 3-trees, Technical Report CIS-TR-86-07, Dept. of Computer and Information Science, University of Oregon. As cited by El-Mallah & Colbourn (1990). Bodlaender, Hans L. (1998), "A partial k-arboretum of graphs with bounded treewidth", Theoretical Computer Science, 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4, hdl:1874/18312, MR 1647486. Dai, Wayne Wei-Ming; Sato, Masao (1990), "Minimal forbidden minor characterization of planar partial 3-trees and application to circuit layout", IEEE International Symposium on Circuits and Systems, vol. 4, pp. 2677–2681, doi:10.1109/ISCAS.1990.112560, S2CID 122926229 El-Mallah, Ehab S.; Colbourn, Charles J. (1990), "On two dual classes of planar graphs", Discrete Mathematics, 80 (1): 21–40, doi:10.1016/0012-365X(90)90293-Q, MR 1045921.
For the characterization in terms of the maximum number of triangles in a planar graph, see Hakimi & Schmeichel (1979). Alon & Caro (1984) quote this result and provide the characterizations in terms of the isomorphism classes of blocks and numbers of blocks. The bound on the total number of cliques follows easily from the bounds on triangles and K₄ subgraphs, and is also stated explicitly by Wood (2007), who provides an Apollonian network as an example showing that this bound is tight. For generalizations of these bounds to nonplanar surfaces, see Dujmović et al. (2009). Hakimi, S. L.; Schmeichel, E. F. (1979), "On the number of cycles of length k in a maximal planar graph", Journal of Graph Theory, 3 (1): 69–86, doi:10.1002/jgt.3190030108, MR 0519175. Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Wood, David R. (2007), "On the maximum number of cliques in a graph", Graphs and Combinatorics, 23 (3): 337–352, arXiv:math/0602191, doi:10.1007/s00373-007-0738-8, MR 2320588, S2CID 46700417. Dujmović, Vida; Fijavž, Gašper; Joret, Gwenaël; Wood, David R. (2009), "The maximum number of cliques in a graph embedded in a surface", European Journal of Combinatorics, 32 (8): 1244–1252, arXiv:0906.4142, Bibcode:2009arXiv0906.4142D, doi:10.1016/j.ejc.2011.04.001, S2CID 1733300.
Zhou et al. (2004); Zhou, Yan & Wang (2005). Zhou, Tao; Yan, Gang; Zhou, Pei-Ling; Fu, Zhong-Qian; Wang, Bing-Hong (2004), Random Apollonian networks, arXiv:cond-mat/0409414v2, Bibcode:2004cond.mat..9414Z. Zhou, Tao; Yan, Gang; Wang, Bing-Hong (2005), "Maximal planar networks with large clustering coefficient and power-law degree distribution", Physical Review E, 71 (4) 046141, arXiv:cond-mat/0412448, Bibcode:2005PhRvE..71d6141Z, doi:10.1103/PhysRevE.71.046141, PMID 15903760, S2CID 21740312.
Albenque & Marckert (2008);Zhang et al. (2008). Zhang, Zhongzhi; Chen, Lichao; Zhou, Shuigeng; Fang, Lujun; Guan, Jihong; Zou, Tao (2008), "Analytical solution of average path length for Apollonian networks", Physical Review E, 77 (1) 017102, arXiv:0706.3491, Bibcode:2008PhRvE..77a7102Z, doi:10.1103/PhysRevE.77.017102, PMID 18351964, S2CID 30404208.
Grünbaum (1963); Grünbaum (1967). Grünbaum, Branko (1963), "Unambiguous polyhedral graphs", Israel Journal of Mathematics, 1 (4): 235–238, doi:10.1007/BF02759726, MR 0185506, S2CID 121075042. Grünbaum, Branko (1967), Convex Polytopes, Wiley Interscience.
Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008). Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Zickfeld, Florian; Ziegler, Günter M. (2006), "Integer realizations of stacked polytopes", Workshop on Geometric and Topological Combinatorics (PDF). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657.

siam.org (Global: low place; English: 6,473^rd place)

Demaine & Schulz (2011). Demaine, Erik; Schulz, André (2011), "Embedding stacked polytopes on a polynomial-size grid", Proc. ACM-SIAM Symposium on Discrete Algorithms (PDF), pp. 1177–1187, archived from the original (PDF) on 2011-06-01, retrieved 2011-03-07.

uah.es (Global: low place; English: low place)

www2.uah.es

Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008). Alon, N.; Caro, Y. (1984), "On the number of subgraphs of prescribed type of planar graphs with a given number of vertices", in Rosenfeld, M.; Zaks, J. (eds.), Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009. Zickfeld, Florian; Ziegler, Günter M. (2006), "Integer realizations of stacked polytopes", Workshop on Geometric and Topological Combinatorics (PDF). Felsner, Stefan; Zickfeld, Florian (2008), "On the number of planar orientations with prescribed degrees" (PDF), Electronic Journal of Combinatorics, 15 (1): R77, arXiv:math/0701771, Bibcode:2007math......1771F, doi:10.37236/801, MR 2411454, S2CID 13893657.

ucsd.edu (Global: 1,933^rd place; English: 1,342^nd place)

math.ucsd.edu

Butler & Graham (2010). Butler, S.; Graham, Ron (2010), "Iterated triangle partitions", in Katona, G.; Schrijver, A.; Szonyi, T. (eds.), Fete of Combinatorics and Computer Science (PDF), Bolyai Society Mathematical Studies, vol. 29, Heidelberg: Springer-Verlag, pp. 23–42.

web.archive.org (Global: 1^st place; English: 1^st place)

Demaine & Schulz (2011). Demaine, Erik; Schulz, André (2011), "Embedding stacked polytopes on a polynomial-size grid", Proc. ACM-SIAM Symposium on Discrete Algorithms (PDF), pp. 1177–1187, archived from the original (PDF) on 2011-06-01, retrieved 2011-03-07.

Apollonian network (English Wikipedia)

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