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==References== {{reflist|refs= <ref name=1look>{{citation | last1 = Clark | first1 = John | last2 = Holton | first2 = Derek Allan | isbn = 9788170234630 | page = 28 | publisher = Allied Publishers | title = A First Look at Graph Theory | url = https://books.google.com/books?id=kb-aLlgYZuEC&pg=PA28 | year = 1995 | access-date = 2022-01-07 | archive-date = 2022-01-08 | archive-url = https://web.archive.org/web/20220108225127/https://books.google.com/books?id=kb-aLlgYZuEC&pg=PA28 | url-status = live }}</ref> <ref name=bengalloun>{{citation | last = Bengelloun | first = Safwan Abdelmajid | date = December 1982 | id = {{ProQuest|303248045}} | page = 12 | publisher = Yale University | title = Aspects of Incremental Computing | type = PhD thesis}}</ref> <ref name=berge>{{citation | last = Berge | first = Claude | author-link = Claude Berge | journal = [[Comptes rendus de l'Académie des Sciences|Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences]] | mr = 100850 | pages = 258–259 | title = Sur le couplage maximum d'un graphe | volume = 247 | year = 1958}}</ref> <ref name=berkramal>{{citation | last1 = Berenbrink | first1 = Petra | last2 = Krayenhoff | first2 = Bruce | last3 = Mallmann-Trenn | first3 = Frederik | doi = 10.1016/j.ipl.2014.05.008 | issue = 11 | journal = [[Information Processing Letters]] | mr = 3230913 | pages = 639–642 | title = Estimating the number of connected components in sublinear time | volume = 114 | year = 2014}}</ref> <ref name=bollobas>{{citation | last = Bollobás | first = Béla | author-link = Béla Bollobás | doi = 10.1007/978-1-4612-0619-4 | isbn = 0-387-98488-7 | mr = 1633290 | page = 6 | publisher = Springer-Verlag | location = New York | series = Graduate Texts in Mathematics | title = Modern Graph Theory | url = https://books.google.com/books?id=JeIlBQAAQBAJ&pg=PA6 | volume = 184 | year = 1998 | access-date = 2022-01-08 | archive-date = 2022-01-08 | archive-url = https://web.archive.org/web/20220108192625/https://books.google.com/books?id=JeIlBQAAQBAJ&pg=PA6 | url-status = live }}</ref> <ref name=boost>{{citation | last1 = Siek | first1 = Jeremy | last2 = Lee | first2 = Lie-Quan | last3 = Lumsdaine | first3 = Andrew | contribution = 7.1 Connected components: Definitions | pages = 97–98 | publisher = Addison-Wesley | title = The Boost Graph Library: User Guide and Reference Manual | year = 2001}}</ref> <ref name=chvatal>{{citation | last = Chvátal | first = Václav | author-link = Václav Chvátal | doi = 10.1016/0012-365X(73)90138-6 | doi-access = free | issue = 3 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]] | mr = 0316301 | pages = 215–228 | title = Tough graphs and Hamiltonian circuits | volume = 5 | year = 1973}}</ref> <ref name=cioaba>{{citation | last = Cioabă | first = Sebastian M. | editor-last = Dehmer | editor-first = Matthias | contribution = Some applications of eigenvalues of graphs | doi = 10.1007/978-0-8176-4789-6_14 | location = New York | mr = 2777924 | pages = 357–379 | publisher = Birkhäuser/Springer | title = Structural Analysis of Complex Networks | year = 2011| isbn = 978-0-8176-4788-9 }}; see [https://books.google.com/books?id=0ymYdW1rtBUC&pg=PA361 proof of Lemma 5, p. 361] {{Webarchive|url=https://web.archive.org/web/20220108230453/https://books.google.com/books?id=0ymYdW1rtBUC&pg=PA361 |date=2022-01-08 }}</ref> <ref name=cohav>{{citation | last1 = Cohen | first1 = Reuven | last2 = Havlin | first2 = Shlomo | contribution = 10.1 Percolation on complex networks: Introduction | contribution-url = https://books.google.com/books?id=1ECLiFrKulIC&pg=PA97 | isbn = 978-1-139-48927-0 | pages = 97–98 | publisher = Cambridge University Press | title = Complex Networks: Structure, Robustness and Function | year = 2010 | access-date = 2022-01-10 | archive-date = 2022-01-10 | archive-url = https://web.archive.org/web/20220110174730/https://books.google.com/books?id=1ECLiFrKulIC&pg=PA97 | url-status = live }}</ref> <ref name=dst>{{citation | last1 = Dillencourt | first1 = Michael B. | last2 = Samet | first2 = Hanan | author2-link = Hanan Samet | last3 = Tamminen | first3 = Markku | doi = 10.1145/128749.128750 | issue = 2 | journal = [[Journal of the ACM]] | mr = 1160258 | pages = 253–280 | title = A general approach to connected-component labeling for arbitrary image representations | volume = 39 | year = 1992| s2cid = 1869184 | citeseerx = 10.1.1.73.8846 }}</ref> <ref name=foldes>{{citation | last = Foldes | first = Stephan | isbn = 978-1-118-03143-8 | page = 199 | publisher = John Wiley & Sons | title = Fundamental Structures of Algebra and Discrete Mathematics | url = https://books.google.com/books?id=A59RivXPMzkC&pg=PA199 | year = 2011 | access-date = 2022-01-07 | archive-date = 2022-01-07 | archive-url = https://web.archive.org/web/20220107080529/https://books.google.com/books?id=A59RivXPMzkC&pg=PA199 | url-status = live }}</ref> <ref name=frikar>{{citation | last1 = Frieze | first1 = Alan | author1-link = Alan M. Frieze | last2 = Karoński | first2 = Michał | contribution = 1.1 Models and relationships | doi = 10.1017/CBO9781316339831 | isbn = 978-1-107-11850-8 | mr = 3675279 | pages = 3–9 | publisher = Cambridge University Press, Cambridge | title = Introduction to Random Graphs | year = 2016}}</ref> <ref name=subcritical>{{harvtxt|Frieze|Karoński|2016}}, 2.1 Sub-critical phase, pp. 20–33; see especially Theorem 2.8, p. 26, Theorem 2.9, p. 28, and Lemma 2.11, p. 29</ref> <ref name=supercritical>{{harvtxt|Frieze|Karoński|2016}}, 2.2 Super-critical phase, pp. 33; see especially Theorem 2.14, p. 33–39</ref> <ref name=critical>{{harvtxt|Frieze|Karoński|2016}}, 2.3 Phase transition, pp. 39–45</ref> <ref name=random-connectivity>{{harvtxt|Frieze|Karoński|2016}}, 4.1 Connectivity, pp. 64–68</ref> <ref name=hhkp>{{citation | last1 = Huang | first1 = Shang-En | last2 = Huang | first2 = Dawei | last3 = Kopelowitz | first3 = Tsvi | last4 = Pettie | first4 = Seth | editor-last = Klein | editor-first = Philip N. | arxiv = 1609.05867 | contribution = Fully dynamic connectivity in <math>O\bigl(\log n(\log\log n)^2\bigr)</math> amortized expected time | doi = 10.1137/1.9781611974782.32 | pages = 510–520 | title = Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19 | year = 2017| s2cid = 15585534 }}</ref> <ref name=hoptar>{{citation | last1 = Hopcroft | first1 = John | author1-link = John Hopcroft | last2 = Tarjan | first2 = Robert | author2-link = Robert Tarjan | date = June 1973 | doi = 10.1145/362248.362272 | issue = 6 | journal = [[Communications of the ACM]] | pages = 372–378 | title = Algorithm 447: efficient algorithms for graph manipulation | volume = 16| s2cid = 14772567 | doi-access = free }}</ref> <ref name=jnp>{{citation | last1 = Joyner | first1 = David | last2 = Nguyen | first2 = Minh Van | last3 = Phillips | first3 = David | contribution = 1.6.1 Union, intersection, and join | contribution-url = https://code.google.com/p/graphbook/ | date = May 10, 2013 | edition = 0.8-r1991 | pages = 34–35 | publisher = Google | title = Algorithmic Graph Theory and Sage | access-date = January 8, 2022 | archive-date = January 16, 2016 | archive-url = https://web.archive.org/web/20160116090036/https://code.google.com/p/graphbook/ | url-status = live }}</ref> <ref name=knuth>{{citation | last = Knuth | first = Donald E. | author-link = Donald Knuth | contribution = Weak components | date = January 15, 2022 | pages = 11–14 | title = The Art of Computer Programming, Volume 4, Pre-Fascicle 12A: Components and Traversal | url = https://cs.stanford.edu/~knuth/fasc12a+.pdf | access-date = March 1, 2022 | archive-date = January 18, 2022 | archive-url = https://web.archive.org/web/20220118183018/https://cs.stanford.edu/~knuth/fasc12a+.pdf | url-status = live }}</ref> <ref name=kozen>{{citation | last = Kozen | first = Dexter C. | author-link = Dexter Kozen | contribution = 4.1 Biconnected components | contribution-url = https://books.google.com/books?id=HFn1BwAAQBAJ&pg=PA20 | doi = 10.1007/978-1-4612-4400-4 | isbn = 0-387-97687-6 | mr = 1139767 | pages = 20–22 | publisher = Springer-Verlag | location = New York | series = Texts and Monographs in Computer Science | title = The Design and Analysis of Algorithms | year = 1992 | s2cid = 27747202 | access-date = 2022-01-08 | archive-date = 2022-01-08 | archive-url = https://web.archive.org/web/20220108192351/https://books.google.com/books?id=HFn1BwAAQBAJ&pg=PA20 | url-status = live }}</ref> <ref name=lewpap>{{citation | last1 = Lewis | first1 = Harry R. | author1-link = Harry R. Lewis | last2 = Papadimitriou | first2 = Christos H. | author2-link = Christos Papadimitriou | doi = 10.1016/0304-3975(82)90058-5 | issue = 2 | journal = [[Theoretical Computer Science (journal)|Theoretical Computer Science]] | mr = 666539 | pages = 161–187 | title = Symmetric space-bounded computation | volume = 19 | year = 1982| doi-access = free }}</ref> <ref name=lewzax>{{citation | last1 = Lewis | first1 = Harry | author1-link = Harry R. Lewis | last2 = Zax | first2 = Rachel | isbn = 978-0-691-19061-7 | page = 145 | publisher = Princeton University Press | title = Essential Discrete Mathematics for Computer Science | url = https://books.google.com/books?id=fAZ-DwAAQBAJ&pg=PA145 | year = 2019 | access-date = 2022-01-08 | archive-date = 2022-01-08 | archive-url = https://web.archive.org/web/20220108191703/https://books.google.com/books?id=fAZ-DwAAQBAJ&pg=PA145 | url-status = live }}</ref> <ref name=mcnosh>{{citation | last1 = McColl | first1 = W. F. | last2 = Noshita | first2 = K. | doi = 10.1016/0166-218X(86)90020-X | issue = 1 | journal = [[Discrete Applied Mathematics]] | mr = 856101 | pages = 67–73 | title = On the number of edges in the transitive closure of a graph | volume = 15 | year = 1986| doi-access = }}</ref> <ref name=read>{{citation | last = Read | first = Ronald C. | author-link = Ronald C. Read | doi = 10.1016/S0021-9800(68)80087-0 | journal = [[Journal of Combinatorial Theory]] | mr = 224505 | pages = 52–71 | title = An introduction to chromatic polynomials | volume = 4 | year = 1968| doi-access = free }}; see Theorem 2, p. 59, and corollary, p. 65</ref> <ref name=reingold>{{citation | last = Reingold | first = Omer | author-link = Omer Reingold | doi = 10.1145/1391289.1391291 | doi-access = | issue = 4 | journal = [[Journal of the ACM]] | mr = 2445014 | page = A17:1–A17:24 | title = Undirected connectivity in log-space | volume = 55 | year = 2008| s2cid = 207168478 }}</ref> <ref name=skiena>{{citation | last = Skiena | first = Steven | author-link = Steven Skiena | contribution = 6.1.2 Kruskal's Algorithm | contribution-url = https://books.google.com/books?id=7XUSn0IKQEgC&pg=PA196 | doi = 10.1007/978-1-84800-070-4 | isbn = 978-1-84800-069-8 | pages = 196–198 | publisher = Springer | title = The Algorithm Design Manual | year = 2008 | bibcode = 2008adm..book.....S | access-date = 2022-01-07 | archive-date = 2022-01-07 | archive-url = https://web.archive.org/web/20220107183740/https://books.google.com/books?id=7XUSn0IKQEgC&pg=PA196 | url-status = live }}</ref> <ref name=thuswa>{{citation | last1 = Thulasiraman | first1 = K. | last2 = Swamy | first2 = M. N. S. | isbn = 978-1-118-03025-7 | page = 9 | publisher = John Wiley & Sons | title = Graphs: Theory and Algorithms | url = https://books.google.com/books?id=rFH7eQffQNkC&pg=PA9 | year = 2011 | access-date = 2022-01-07 | archive-date = 2022-01-07 | archive-url = https://web.archive.org/web/20220107215533/https://books.google.com/books?id=rFH7eQffQNkC&pg=PA9 | url-status = live }}</ref> <ref name=tutte-betti>{{citation | last = Tutte | first = W. T. | author-link = W. T. Tutte | isbn = 0-201-13520-5 | mr = 746795 | page = 15 | publisher = Addison-Wesley | location = Reading, Massachusetts | series = Encyclopedia of Mathematics and its Applications | title = Graph Theory | url = https://books.google.com/books?id=uTGhooU37h4C&pg=PA15 | volume = 21 | year = 1984 | access-date = 2022-01-07 | archive-date = 2022-01-07 | archive-url = https://web.archive.org/web/20220107081347/https://books.google.com/books?id=uTGhooU37h4C&pg=PA15 | url-status = live }}</ref> <ref name=tutte-matching>{{citation | last = Tutte | first = W. T. | author-link = W. T. Tutte | doi = 10.1112/jlms/s1-22.2.107 | journal = [[The Journal of the London Mathematical Society]] | mr = 23048 | pages = 107–111 | title = The factorization of linear graphs | volume = 22 | year = 1947| issue = 2 }}</ref> <ref name=wilson>{{citation | last = Wilson | first = R. J. | author-link = Robin Wilson (mathematician) | doi = 10.1080/00029890.1973.11993318 | journal = [[The American Mathematical Monthly]] | jstor = 2319608 | mr = 371694 | pages = 500–525 | title = An introduction to matroid theory | volume = 80 | year = 1973| issue = 5 }}</ref> <ref name=wood>{{citation | last = Wood | first = David R. | author-link = David Wood (mathematician) | editor-last = Kao | editor-first = Ming-Yang | contribution = Three-dimensional graph drawing | doi = 10.1007/978-3-642-27848-8_656-1 | pages = 1–7 | publisher = Springer | title = Encyclopedia of Algorithms | url = https://users.monash.edu/~davidwo/papers/EncPaper.pdf | year = 2014 | isbn = 978-3-642-27848-8 | access-date = 2022-01-08 | archive-date = 2022-01-08 | archive-url = https://web.archive.org/web/20220108225308/https://users.monash.edu/~davidwo/papers/EncPaper.pdf | url-status = live }}</ref> <ref name=wulff-nilsen>{{citation | last = Wulff-Nilsen | first = Christian | editor-last = Khanna | editor-first = Sanjeev | arxiv = 1209.5608 | contribution = Faster deterministic fully-dynamic graph connectivity | doi = 10.1137/1.9781611973105.126 | pages = 1757–1769 | title = Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013 | year = 2013| isbn = 978-1-61197-251-1 | s2cid = 13397958 }}</ref> }}
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