81. Properties Of Topological Spaces it contains cl(cl(A)). Remark These four properties are sometimes called the kuratowskiaxioms after the Polish mathematician kazimierz kuratowski (1896 to 1980 http://www.gap-system.org/~john/MT3822/Lectures/L12.html

Metric and Topological Spaces Previous page (Definition and examples of topologies) Contents Next page (Continuity for topological spaces) Properties of topological spaces We can recover some of the things we did for metric spaces earlier. Definition A subset A of a topological space X is called closed if X A is open in X Then closed sets satisfy the following properties and X are closed A B closed A B is closed A i i I A i is closed Proof Take complements. So the set of all closed sets is closed [!] under finite unions and arbitrary intersections. As in the metric space case, we have Definition A point x is a limit point of a set A if every open set containing x meets A (in a point x Theorem A set is closed if and only if it contains all its limit points Proof Imitate the metric space proof. Definitions The interior int A ) of a set A is the largest open set A The closure cl A ) of a set A is the smallest closed set containing A It is easy to see that int A ) is the union of all the open sets of X contained in A and cl A ) is the intersection of all the closed sets of X containing A Some properties cl A cl A ) for any subset A cl A B cl A cl B ) for any subsets A and B cl cl A cl A ) for any subset A Proof K1. and K2. follow from the definition.

82. Carl McTague This is a fun, relatively wellknown problem credited to kazimierz kuratowski(1896-1980). I was really taken by the problem when I first read it. http://www.mctague.org/carl/

Cincinnati, May 5, 2002 Photo from the Pie Shoot by Colleen Carl McTague Welcome! "Hear that? It's one of the most exciting sounds in the world... bongo!" Hi there! If you can't already tell, I'm a big dork who just turned 22 - way too old. I study math and music in Cincinnati ( USA ), where my family has lived for the last few years (much too long in my opinion), but spend most of my free time in Santa Fe doing research. Someday I'll grow up and go to grad school somewhere fun (I hope). I throw up little documents here once in a while that I think are cool: like formal toys for composing music through computation, nifty things to do with the Axiom of Choice, eventually pattern formation in spatially extended systems ( "McTaag, are you playing with your little machines again?" -Homayoun Bagheri), giving my little sis' a desperately needed lemon-cream pie in the face full series )...oh yeah, and recordings and descriptions of music I've composed! Tabula Profunda "Be still Taggart, be still! My mind is aglow with whirling, transient nodes of thought...careening through a cosmic vapor of invention!" -Hedley Lamarr A Brief on the Xi Operator May 2002, edited November 2002

83. Philosophy firstorder logic. Thales of Miletus; Pythagoras of Samos; Ptolemy;kazimierz kuratowski; Kurt G÷del; Benoit Mandelbrot; Witold Hurewicz; http://www.geo.ryerson.ca/~michalak/html/philo.html

Philosophy and Mathematics Ernest Gellner Principia Cybernetica Project The Project's aim is the computer-supported collaborative development of an evolutionary-systemic philosophy. Put more simply, PCP tries to tackle age-old philosophical questions with the help of the most recent cybernetic theories and technologies. Analytic Philosophy Forum a public and moderated mailing list for the discussion of analytic philosophy, its history, its literature, and other topics of related interest. Fuzzy Systems a tutorial by James F. Brule History of Mathematics and Mathematical Logic Alfred Tarski Logic Software from CSLI Tarski's World introduces the language of first-order logic. Thales of Miletus Pythagoras of Samos Ptolemy Kazimierz Kuratowski ... Andrzej Mostowski Return to home page

84. Seminar-Workshop Bibliography kuratowski, kazimierz, Fifty Years of Polish Mathematics Remembrancesand Reflections. New York Warszawa Pergamon Press/ Polish http://thisisnotthat.com/gs/rp_gsbib.html

The Dallas-Fort Worth Center for General Semantics 1412 Texas Street, P.O. Box 1565 Fort Worth, TX 76101-1565 untangle the tangled webs you verbally weave General-Semantics Seminar-Workshop Bibliography Revised, Updated and Annotated (1995) by Robert P. Pula (Provided courtesy of the Institute of General Semantics General-semantics, begun with Korzybski's definition of humans as time-binders in 1921, presented as a system/discipline in 1933, has, in the last six decades, become a field. A public-sized library of books, articles, papers, studies, dissertations even a few notorious novels rest on shelves all over the planet. Some of them are excellent. Given the foundations of general-semantics, 'its' library needs to include writings from related fields. " So many books, so little time ." Our seminar bibliography, then, must be highly selective, limited to items that for historical as well as formulational reasons I deem required reading for the well-informed, well-trained, understanding general-semanticist.

85. Polska Szko³a Matematyczna Tacy mlodzi studenci jak Bronislaw Knaster, Stanislaw Saks, Antoni Zygmund,kazimierz kuratowski, Alfred Tarski, kazimierz Zarankiewicz, osiagneli http://www.mt.com.pl/num/09_00/matma.htm

M³ody Technik Polska Szko³a Matematyczna Wrzesieñ 2000 Nowy rok szkolny przywitajmy tym razem rozwa¿aniami dotycz±cymi rozwoju polskiej my¶li matematycznej, jej rodowodu oraz znaczenia dla innych nauk. Profesor Stefan Banach "Polska eksportuje wêgiel i twierdzenia matematyczne" powiedzia³ w 1946 roku Stanis³aw Skrzeszewski, ówczesny dyrektor departamentu w Ministerstwie O¶wiaty. Istotnie, wêgiel by³ wtedy niemal jedynym bogactwem materialnym, który mogli¶my eksportowaæ, a matematyka jednym z niewielu dóbr duchowych, z których Polska lat miêdzywojennych by³a s³ynna na ca³y ¶wiat. ¯eby zrozumieæ fenomen Polskiej Szko³y Matematycznej, trzeba cofn±æ siê do lat po powstaniu styczniowym. Po kolejnym przegranym powstaniu do g³osu doszli ludzie, o których mówi siê, ¿e byli nudni i ma³o romantyczni. To pozytywi¶ci. To oni g³osili, ¿e zamiast organizowaæ kolejne zrywy przeciw zaborcom, nale¿y po prostu rozwijaæ naukê, gospodarkê, sztukê i technikê. Dbaæ o polsk± kulturê - na ile to mo¿liwe w niewoli. A wtedy niepodleg³o¶æ bêdzie ³atwiej wywalczyæ i ³atwiej utrzymaæ. Jedn± z takich pozytywistycznych instytucji by³a Kasa im. Mianowskiego, patronuj±ca nauce i naukowcom na ziemiach polskich. Powsta³a ona w 1881 roku. Wydawano ksi±¿ki, wysy³ano m³odych uczonych za granicê, a w kraju organizowano kursy i nieformalne uniwersytety. Do takich nazw instytucji powo³anych przez kasê im. Mianowskiego, jak Uniwersytet Lataj±cy i Towarzystwo Kursów Naukowych, nawi±zywa³a opozycja w Polsce w latach siedemdziesi±tych i osiemdziesi±tych XX wieku.

86. Autores 1870 - 1938.(1); kuratowski, kazimierz.(1); kuratowski, Kazinierz http://biblioteca.ipb.upel.edu.ve/ALEXANDR/CATALOGOS/pbqmto/Cat.Aut_11.HTM

K k., Flower Khn, Fritz. Khnemann, Ursula. Kac, ark. ... Kvaraceus, William C.

87. URANOS: Polskie Nazwy W Kosmosie kazimierz kuratowski (18961980), polski matematyk (teoria mnogosci,topologia), wiceprezes Miedzynarodowej Unii Matematycznej. http://www.uranos.eu.org/poland/plnames.html

Polskie nazwy w kosmosie UWAGA : zapraszamy do podpisywania internetowej petycji w sprawie wys³ania na Marsa pojazdu "Marie Curie"! Podajemy tutaj, mamy nadziejê pe³n±, listê obiektów pozaziemskich (w tym sztucznych, np. satelitów) z nazwami polskimi lub maj±cymi istotny zwi±zek z Polsk±, wraz z krótkimi informacjami o ich ¼ród³ach. Dane astronomiczne obiektow pochodz± w wiêkszo¶ci z internetowego katalogu nazw IAU (Miêdzynarodowej Unii Astronomicznej) oraz z katalogu danych planetoid na witrynie NASA Solar System Dynamics Group . Polsko¶æ nazwy definiujemy nieco szerzej, ni¿ w katalogu nazw IAU , z którego pochodzi wiele zamieszczonych tu nazw (oprócz nazw planetoid, komet i obiektów sztucznych). W katalogu tym ka¿da nazwa ma przyznan± pojedyncz± atrybucjê narodow±, tak¿e w wypadkach, gdy nazwa pochodzi od obiektu (np. osoby), do której mo¿e sobie ro¶ciæ pretensje kilka narodowo¶ci: np. Cio³kowski - Rosjanin polskiego pochodzenia - "przyznany" jest w katalogu tylko Rosji, za¶ Mariê Goeppert-Mayer , niemieck± uczon± pracuj±c± w USA, przyznano nieoczekiwanie Polsce (jej zwi±zki z Polska ograniczaj± siê tylko do urodzenia w Katowicach, sk±d wyjecha³a do Niemiec jako ma³e dziecko, wraz ze sw± niemieck± rodzin±). W niektórych przypadkach katalog podaje szersz± atrybucjê, w której jednak mie¶ci siê Polska tak¿e (np. nazwê £ada przyznano ogólnie "europejskim S³owianom"). Dodali¶my tak¿e niektóre nazwy, których zwi±zek z Polsk± jest po¶redni (np. kratery i planetoida

88. Math 305 Gallery Of Mathematicians Christian Goldbach 16901764, Pafnuty Chebyshev 1821-1894 S 1, René Descartes1596-1650 S 1 S 2, Norbert Wiener 1894-1964, kazimierz kuratowski 1896-1980. http://www.math.umt.edu/~stroet/305Gallery.html

Math 305 Gallery of Mathematicians Following is a gallery of small portraits of mathematicians as they made their appearance in the course. Click on the portrait to get a poster or larger picture; click on the name to get a biography. If available, click on S to get a stamp related to the mathematician. Charles Dodgson Augustus De Morgan Benjamin Peirce Joseph-Louis Lagrange ... Euclid of Alexandria c.325BC-c.265BC Pythagoras of Samos c.569BC-c.475BC S S S Diophantus of Alexandria c.200-c.284 Andrew John Wiles Georg Cantor John Venn Venn diagrams ... Paul Cohen

89. Kazimierz Kuratowski - Wikipedia Similar pages Book review Handbook of the History of General Topology, Volume The individual contributions and influences of Waclaw Sierspinski (18821969) andof kazimierz kuratowski (1896-1980), are discussed by Ryszard Engelking, and R http://www.wikipedia.org/wiki/Kuratowski%27s_theorem

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles Interlanguage links All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Other languages: Deutsch Polski Kazimierz Kuratowski (Redirected from Kuratowski's theorem Kazimierz Kuratowski (born February 2 Warsaw , died June 18 Warsaw ) was a Polish mathematician Among his contributions to mathematics are: a characterization of Hausdorff spaces in terms of what are now called Kuratowski closure operators the proof that a graph with no vertices of order 2 is nonplanar if and only if it contains a copy of K (the full graph on 5 vertices) or K (six vertices, three of which connect to each of the other three, for a total of nine edges). External links: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kuratowski.html A biography of Kuratowski. Edit this page Discuss this page Older versions What links here ... Related changes Other languages: Deutsch Polski Main Page About Wikipedia ... Recent changes It was last modified 06:42 Sep 20, 2002. All text is available under the terms of the

90. Matematycy Polscy - Matematyka - Wirtualny Wszech¶wiat ZARANKIEWICZ kazimierz (19021959). Urodzil sie 2 maja 1902 w Czestochowie.W 1919 ukonczyl gimnazjum w Bedzinie, nastepnie http://www.wiw.pl/matematyka/Biogramy/Biogramy_22.Asp

W iw.pl Na bie¿±co: I nformacje C o nowego Matematyka i przyroda: A stronomia B iologia ... odelowanie rzeczywisto¶ci Humanistyka: F ilozofia H istoria ... ztuka Czytaj: B iblioteka D elta ... ielcy i wiêksi Przydatne: S ³owniki C o i gdzie studiowaæ ... szech¶wiat w obrazkach Jeste¶ tutaj: Wirtualny Wszech¶wiat Matematyka Matematycy polscy Jeste¶ tutaj Matematycy polscy Spis rzeczy Indeks Auerbach Herman Banach Stefan Borsuk Karol ... Zarankiewicz Kazimierz Szukacz Przeszukaj Wirtualny Wszech¶wiat: Jak zadawaæ pytania? ZARANKIEWICZ KAZIMIERZ Urodzi³ siê 2 maja 1902 w Czêstochowie. W 1919 ukoñczy³ gimnazjum w Bêdzinie, nastêpnie studiowa³ matematykê na Uniwersytecie Warszawskim, uzyskuj±c w 1923 stopieñ doktora filozofii na podstawie pracy Sur les points de division dans les ensembles connexes ("Fundamenta Mathematicae", 9/1927). W 1929 habilitowa³ siê na podstawie pracy Über eine topologische Eingenschaft der Ebene Matematyka wy¿sza dla studentów wy¿szych szkó³ le¶nych i rolniczych (skrypt 1952) i Mechanika teoretyczna (Warszawa 1955). Interesowa³ siê astronautyk±, by³ organizatorem i pierwszym prezesem Polskiego Towarzystwa Astronautycznego. Zmar³ nagle 5 listopada 1959, przewodnicz±c posiedzeniu plenarnemu na X Zje¼dzie Miêdzynarodowej Federacji Astronautycznej w Londynie.

91. Biography-center - Letter K Kuperberg, Krystyna www.agnesscott.edu/lriddle/women/kuper.htm; kuratowski, Kazimierzwwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/kuratowski.html; http://www.biography-center.com/k.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish K 374 biographies www-history.mcs.st-and.ac.uk/~history/Mathematicians/Konig_Julius.html www-history.mcs.st-and.ac.uk/~history/Mathematicians/Konig_Samuel.html www-history.mcs.st-and.ac.uk/~history/Mathematicians/Konigsberger.html www-history.mcs.st-and.ac.uk/~history/Mathematicians/Kurschak.html Kabir, www.geocities.com/athens/8107/bios1.html#kabir Kac, Mark www-history.mcs.st-and.ac.uk/~history/Mathematicians/Kac.html Kaestner, Abraham www-history.mcs.st-and.ac.uk/~history/Mathematicians/Kaestner.html Kagan, Benjamin www-history.mcs.st-and.ac.uk/~history/Mathematicians/Kagan.html Kahanamoku, Duke Paoa www.olympic.org/uk/athletes/heroes/bio_uk.asp?PAR_I_ID=54152 Kahlbaum, Karl Ludwig www.whonamedit.com/doctor.cfm/624.html Kahler, Otto

92. The Legacy Of R. L. Moore - Moore, Robert L. -- Center For American History User http://www.discovery.utexas.edu/rlm/reference/cah.html

Moore, Robert L. Center for American History User's Guide Moore, R.L. (Robert Lee), 1882- TITLE: Moore, R.L., Papers, 1898-1974. DESCRIPTION: 32 ft. NOTES: Organized into four series: 1. Mathematical papers. 2. Correspondence. 3. University of Texas, Teaching, National Academy of Sciences. 4. Personal. Summary: Collection documents the career of R.L. Moore (1882-1974) at the University of Texas (1920-1974), with a small amount of material concerning his doctoral studies at the University of Chicago. The papers reflect Moore's research in point-set topology. There are records of Moore's presidency of the American Mathematical Society (1937-39). The papers also include a collection of G.B. Halsted's articles and translations, together with publications about Halsted. Reprints of Moore's papers, Moore's reprint collection, and theses and dissertations prepared under his supervision are included. Correspondents include R.C. Archibald, S. Armentrout, J. and L. Barrett, E.F. Beckenbach, E.T. Bell, R.H. Bing, G.D. and G. Birkhoff, G.A. Bliss, M. Bocher, E.W. Chittenden, L.E. Dickson, E. Dyer, M. Frechet, G.B. Halsted, J.R. Kline, C. Kuratowski, S. Lefschetz, E.H. Moore, R.G.D. Richardson, M.E. Rudin, W. Sierpinski, J.M. Slye, M. Stone, O. Veblen, G.T. Whyburn, and R.L. Wilder. Material includes correspondence, research notebooks, drafts, teaching material, reprints, photographs, and sound recordings. Before 1984 held by the University of Texas at Austin Humanities Research Center.

93. Katalogi - Katalogi BGPP - Wykaz Podrêczników - Automatyka I Robotyka Warszawa, 1989. Przedmiot LOGIKA MATEMATYCZNA I TEORIA MNOGOSCI KuratowskiKazimierz, Wstep do teorii mnogosci i topologii. Warszawa http://www.ml.put.poznan.pl/katalogi/p_podrecznik_el_aut.shtml

Wstecz WYDZIA£ ELEKTRYCZNY Kierunek: AUTOMATYKA I ROBOTYKA Rok 1 Rok 2 Rok 3 Rok 4 ... Rok 5 ROK 1 Przedmiot: ANALIZA MATEMATYCZNA ¯akowski W. , Matematyka. T. I, II, III. Warszawa, WNT, 1970 Krysicki W. , W³odarski L., Analiza matematyczna w zadaniach. T. I, II. Warszawa, PWN, 1983 Stankiewicz W. , Zadania z matematyki. T. I, II. Warszawa, PWN, 1976 Przedmiot: MECHANIKA I WYTRZYMA£O¦Æ MATERIA£ÓW Leyko J. , Mechanika ogólna. T. I. II. Warszawa, PWN Mieszczerski W. , Zbiór zadañ z mechaniki. Warszawa, PWN, 1970 Niezgodziñski , Niezgodziñski, Wytrzyma³o¶æ materia³ów. Warszawa, PWN, 1998 Banasiak , Grossman, Trombski, Zbiór zadañ z wytrzyma³o¶ci materia³ów. Warszawa, PWN, 1998 Przedmiot: ALGEBRA I WYBRANE DZIA£Y GEOMETRII Goetz A. , Geometria ró¿niczkowa. Mostowski Andrzej , Stark Marceli, Algebra liniowa. Wyd. 4. Warszawa, PWN, 1975 Stark Marceli , Geometria analityczna z wstêpem do geometrii wielowymiarowej. Wyd. 4. Warszawa, PWN, 1970 Mostowski Andrzej , Stark Marceli, Algebra wy¿sza. Wyd. 2. Warszawa, PWN, 1966 Przedmiot: FIZYKA Halliday David , Resnik Robert, Fizyka dla studentów nauk przyrodniczych i technicznych. Wyd. 4. Warszawa, PWN, 1972

94. Pronunciation Guide To Mathematicians The summary for this Korean page contains characters that cannot be correctly displayed in this language/character set. http://puzzle.jmath.net/math/mathcian/

Pronunciation Guide to Mathematicians:¼öÇÐÀÚÀÇ À̸§ Àбâ Home Math Puzzles Links Contents A B C D ... Abel , Niels Henrik ´Ò½º Ç ¾Æº§ Ackermann , Wilhelm ºôÇ︧ ¾ÆÄ¿¸¸ Agnesi , Maria Gaetana ¸¶¸®¾Æ °¡¿¡Å¸³ª ¾Æ³éÁö Ahlfors , Lars Valerian ¶ó¸£½º ¹ß·¹¸®¾È ¾ËÆ÷¸£½º Ampere , Andre Marie ¾Óµå·¹ ¸¶¸® ¾ÓÆ丣 Argand , Jean Robert Àå ·Îº£¸£ ¾Æ¸£°­ Top B Babbage , Charls Âû½º ¹è¹èÁö Bachet de Meziriac, Claude Gaspard Ŭ·Îµå °¡½ºÆĸ£ ¹Ù¼Î µå ¸ÞÁö¸®¾Ç Baire , Rene Louis ¸£³× ·çÀÌ º£¸£ Banach , Stefan ½ºÅ×ÆÇ ¹Ù³ªÈå Barbier , Joseph Emile Á¶Á¦ÇÁ ¿¡¹Ð ¹Ù¸£ºñ¿¡ Barrow , Issac ¾ÆÀÌÀÛ ¹è·Î Bayes , Thomas Å丶½º º£ÀÌÁî Beltrami , Eugenio ¿¡¿ìÁ¦´Ï¿À º§Æ®¶ó¹Ì Bernoulli , Daniel ´Ù´Ï¿¤ º£¸£´­¸®(º£¸£´©ÀÌ) Bernstein , Felix Æ縯½º º£¸¥½´Å¸ÀÎ Bertrand , Joseph Louis Francois Á¶Á¦ÇÁ ·çÀÌ ÇÁ¶û¼ö¾Æ º£¸£Æ®¶û Besicovitch , Abram Samoilovitch ¾Æºê¶÷ »ç¸ðÀϷκñÄ¡ º£½ÄÚºñÄ¡ Betti , Enrico ¿£¸®ÄÚ º£Æ¼ Bezout , Etienne ¿¡Æ¼¿£ º£ÁÖ Bianchi , Luigi ·çÀÌÁö ºñ¾ÈÅ° Bieberbach , Ludwig Georg Elias ·çÆ®ºñÈ÷ °Ô¿À¸£Å© ¿¤¸®¾Æ½º ºñ¹ö¹ÙÈå Binet , Jacques Philippe Marie ÀÚÅ© Çʸ³ ¸¶¸® ºñ³× Birkhoff , George David Á¶Áö µ¥À̺ñµå ¹öÄÚÇÁ Blichfeldt , Hans Frederick Çѽº ÇÁ·¹µ¥¸¯ ºí¸®È÷ÆçÆ®(ºí¸¯ÆçÆ®) Bolyai , Farkas Wolfgan/Janos Æĸ£Ä«½ º¼ÇÁ°­/¾ß³ë½ º¸¿©ÀÌ (º¼¸®¾ÆÀÌ) Bolzano , Bernhard Placidus Johann Nepomuk º£¸¥Çϸ£Æ® ǶóÄ¡µÎ½º ¿äÇÑ ³×Æ÷¹«Å© º¼Â÷³ë Bonnet , Pierre Ossian ÇÇ¿¡¸£ ¿À½¾Ó º¸³× Boole , George Á¶Áö ºÒ Borel , Felix Edouard Justin Emile Æ縯½º ¿¡µÎ¾Æ¸£ Á㽺ÅÊ ¿¡¹Ð º¸·¼ Borsuk , Karol Ä«·Ñ º¸¸£¼÷ Bourbaki, Nicolas ´ÏÄݶó ºÎ¸£¹ÙÅ°

95. Citations: Sur Le Probl`eme Des Courbes Gauches En Topologie - Kuratowski (Resea A subdivision of K 5 or K 3,3 that is contained as a . KazimierzKuratowski. Sur le probleme des courbes gauches en topologie. http://citeseer.nj.nec.com/context/353024/0

33 citations found. Retrieving documents... K. Kuratowski. Sur le probleme des courbes gauches en topologie . Fundamenta Mathematicae, 15:271283, 1930. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: The Communication Complexity of Enumeration.. - Ambainis.. (Correct) Here are three examples of sets of graphs closed under minor (g and k are constants) PLANAR = G is Planar GENUS g = G has genus g V C k = G has a vertex cover of size k 1. For PLANAR it is known that the obstruction set is K 5 , K 3,3 (this is not Kuratowski s theorem , that a graph is nonplanar i# it does not have K 5 or K 3,3 as a homeomorphic subgraph, but is easily derivable from it) For the other sets in the example the only proof that there is an obstruction set comes from the Lemma 4. Let H be a fixed graph. It is known [46] that testing if H G .... K. Kuratowski. Sur le probleme des courbes gauches en topologie . Fundamenta Mathematicae, 15:271283, 1930. The Communication Complexity of Enumeration.. - Ambainis.. (Correct) ....are three examples of sets of graphs closed under minor (g and k are constants) PLANAR = G G is Planar GENUS g = G G has genus g V C k = G G has a vertex cover of size k 1. For PLANAR it is known that the obstruction set is K 5 , K 3,3 (this is not Kuratowski s theorem

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