Kuratowski's theorem

A planar graph is a graph whose vertices can be represented by points in the Euclidean plane, and whose edges can be represented by simple curves in the same plane connecting the points representing their endpoints, such that no two curves intersect except at a common endpoint. Planar graphs are often drawn with straight line segments representing their edges, but by Fáry's theorem this makes no difference to their graph-theoretic characterization.

A subdivision of a graph is a graph formed by subdividing its edges into paths of one or more edges. Kuratowski's theorem states that a finite graph $G$ is planar if it is not possible to subdivide the edges of $K\_5$ or $K\_\{3,3\}$, and then possibly add additional edges and vertices, to form a graph isomorphic to $G$. Equivalently, a finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to $K\_5$ or $K\_\{3,3\}$.

{{tesseract_graph_nonplanar_visual_proof.svg}}

If $G$ is a graph that contains a subgraph $H$ that is a subdivision of $K\_5$ or $K\_\{3,3\}$, then $H$ is known as a Kuratowski subgraph of $G$.{{citation

| last = Tutte | first = W. T. | author-link = W. T. Tutte

| journal = Proceedings of the London Mathematical Society

| mr = 0158387

| pages = 743–767

| series = Third Series

| title = How to draw a graph

| volume = 13

| year = 1963

| doi=10.1112/plms/s3-13.1.743}}. With this notation, Kuratowski's theorem can be expressed succinctly: a graph is planar if and only if it does not have a Kuratowski subgraph.

The two graphs $K\_5$ and $K\_\{3,3\}$ are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula. Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph $G$ has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of $G$ itself. Therefore, a graph that contains a Kuratowski subgraph cannot be planar. The more difficult direction in proving Kuratowski's theorem is to show that, if a graph is nonplanar, it must contain a Kuratowski subgraph.

A Kuratowski subgraph of a nonplanar graph can be found in linear time, as measured by the size of the input graph.{{citation

| last = Williamson | first = S. G.

| date = September 1984

| doi = 10.1145/1634.322451

| issue = 4

| journal = J. ACM

| pages = 681–693

| title = Depth-first search and Kuratowski subgraphs

| volume = 31| s2cid = 8348222

| doi-access = free

}}. This allows the correctness of a planarity testing algorithm to be verified for nonplanar inputs, as it is straightforward to test whether a given subgraph is or is not a Kuratowski subgraph.{{citation|title=LEDA: A Platform for Combinatorial and Geometric Computing|first1=Kurt|last1=Mehlhorn|author1-link=Kurt Mehlhorn|first2=Stefan|last2=Näher|page=510|url=https://books.google.com/books?id=Q2aXZl3fgvMC&pg=PA510|publisher=Cambridge University Press|year=1999|isbn=9780521563291}}.

Usually, non-planar graphs contain a large number of Kuratowski-subgraphs. The extraction of these subgraphs is needed, e.g., in branch and cut algorithms for crossing minimization. It is possible to extract a large number of Kuratowski subgraphs in time dependent on their total size.{{citation

| last1 = Chimani| first1 = Markus

| last2 = Mutzel| first2 = Petra | author2-link = Petra Mutzel

| last3 = Schmidt| first3 = Jens M.

| editor1-last = Hong | editor1-first = Seok-Hee | editor1-link = Seok-Hee Hong

| editor2-last = Nishizeki | editor2-first = Takao | editor2-link = Takao Nishizeki

| editor3-last = Quan | editor3-first = Wu

| contribution = Efficient extraction of multiple Kuratowski subdivisions

| isbn = 978-3-540-77536-2

| series = Lecture Notes in Computer Science

| title = Graph Drawing: 15th International Symposium, GD 2007, Sydney, Australia, September 24-26, 2007, Revised Papers

| title-link = International Symposium on Graph Drawing

| date = 2007

| doi = 10.1007/978-3-540-77537-9_17 | doi-access = free

| publisher = Springer

| pages = 159–170

| volume = 4875}}

Kazimierz Kuratowski published his theorem in 1930.{{citation|first=Kazimierz|last=Kuratowski|author-link=Kazimierz Kuratowski|year=1930|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15126.pdf|title=Sur le problème des courbes gauches en topologie|journal=Fund. Math.|volume=15|pages=271–283|doi=10.4064/fm-15-1-271-283 |language=French}}. The theorem was independently proved by Orrin Frink and Paul Smith, also in 1930,{{citation

| last1 = Frink | first1 = Orrin | author-link1 = Orrin Frink

| last2 = Smith | first2 = Paul A. | author-link2 = Paul Althaus Smith

| title = Irreducible non-planar graphs

| journal = Bulletin of the AMS

| volume = 36

| pages = 214

| year = 1930 }} but their proof was never published. The special case of cubic planar graphs (for which the only minimal forbidden subgraph is $K\_\{3,3\}$) was also independently proved by Karl Menger in 1930.{{citation

| last = Menger | first = Karl | author-link = Karl Menger

| title = Über plättbare Dreiergraphen und Potenzen nichtplättbarer Graphen

| journal = Anzeiger der Akademie der Wissenschaften in Wien

| volume = 67

| pages = 85–86

| year = 1930}}

Since then, several new proofs of the theorem have been discovered.{{citation

| last = Thomassen | first = Carsten | author-link = Carsten Thomassen (mathematician)

| doi = 10.1002/jgt.3190050304

| issue = 3

| journal = Journal of Graph Theory

| mr = 625064

| pages = 225–241

| title = Kuratowski's theorem

| volume = 5

| year = 1981}}.

In the Soviet Union, Kuratowski's theorem was known as either the Pontryagin–Kuratowski theorem or the Kuratowski–Pontryagin theorem,{{citation

| last1 = Burstein | first1 = Michael

| doi = 10.1016/0095-8956(78)90024-2

| title = Kuratowski-Pontrjagin theorem on planar graphs

| journal = Journal of Combinatorial Theory, Series B

| volume = 24

| pages = 228–232

| year = 1978| issue = 2

| doi-access =

}}

as the theorem was reportedly proved independently by Lev Pontryagin around 1927.{{citation

| last1 = Kennedy | first1 = John W.

| last2 = Quintas | first2 = Louis V.

| last3 = Sysło | first3 = Maciej M.

| doi = 10.1016/0315-0860(85)90045-X

| title = The theorem on planar graphs

| journal = Historia Mathematica

| volume = 12

| pages = 356–368

| year = 1985| issue = 4

| doi-access = free

}}

However, as Pontryagin never published his proof, this usage has not spread to other places.{{citation|title=Graphs & Digraphs|edition=5th|first1=Gary|last1=Chartrand|author1-link=Gary Chartrand|first2=Linda|last2=Lesniak|first3=Ping|last3=Zhang|author3-link=Ping Zhang (graph theorist)|publisher=CRC Press|year=2010|isbn=9781439826270|page=237|url=https://books.google.com/books?id=K6-FvXRlKsQC&pg=PA237}}.

A closely related result, Wagner's theorem, characterizes the planar graphs by their minors in terms of the same two forbidden graphs $K\_5$ and $K\_\{3,3\}$. Every Kuratowski subgraph is a special case of a minor of the same type, and while the reverse is not true, it is not difficult to find a Kuratowski subgraph (of one type or the other) from one of these two forbidden minors; therefore, these two theorems are equivalent.{{citation|title=Graph Theory|volume=244|series=Graduate Texts in Mathematics|first1=J. A.|last1=Bondy|author1-link=John Adrian Bondy|first2=U.S.R.|last2=Murty|author2-link=U. S. R. Murty|publisher=Springer|year=2008|isbn=9781846289699|page=269|url=https://books.google.com/books?id=HuDFMwZOwcsC&pg=PA269}}.

An extension is the Robertson–Seymour theorem.

• Kelmans–Seymour conjecture, that 5-connected nonplanar graphs contain a subdivision of $K\_5$

{{reflist}}

Category:Statements about planar graphs

Category:Theorems in graph theory